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which subtends an angle radian at the center is equal to r where r is the radius of the circle. Calculating a circle's arc length, central angle, and circumference are not just tasks, but essential skills for geometry, trigonometry and beyond. CHAPTER 5A Central Angles, Arc Length, and Sector Area ... for a central angle of a circle Calculate the arc length and the area of a sector formed by a 30 central The distance along the arc (part of the circumference of a circle, or of any curve). Question: Find the arc length of the circle given by $x^2+y^2=a^2$. It depends on the radius of a circle and the central angle. Geometry Teachers Never Spend Time Trying to Find Materials for Your Lessons Again! Relate the length of an arc to the circumference of a whole circle and the central angle subtended by the arc. The circumference of a circle is an arc measuring 360o. Step 1 : Here, radius = 7cm central angle= 30 degrees. We dare you to prove us wrong. How to Find Arc Length. ... the arc length of a circumscribed circle is: Arc length is a linear measure of the arc measured along the circle. Thus, the length of the arc AB will be 5/18 of the circumference of the circle, which equals 2r, according to the formula for circumference. The relationship of arc length to a ... as the length x of an arc of the unit circle. Join Our Geometry Teacher Community Today! * An alternative definition is that it is an open arc. a. arc length of AB b. circumference of Z c. m RS A connected section of the circumference of a circle. Here's how to calculate the circumference, radius, diameter, arc length and degrees, sector areas, inscribed angles, and other shapes of the circle. and l stand for arc length. Our arc length calculator can calculate the length of an arc of a circle and the area of a sector. Learn how to find the arc with our lesson and try out our examples questions. Geometry calculator solving for circle arc length given radius and central angle An arc is any portion of the	{ "domain": "co.uk", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9865717452580314, "lm_q1q2_score": 0.8964167504529535, "lm_q2_score": 0.9086179031191509, "openwebmath_perplexity": 291.45016114516045, "openwebmath_score": 0.90330970287323, "tags": null, "url": "http://stewswebsolutions.co.uk/journal/gz9p0.php?b2be3c=how-to-find-arc-length-of-a-circle" }
calculator solving for circle arc length given radius and central angle An arc is any portion of the circumference of a circle.http://www.mathwords.com/a/arc_circle.htm Arc length is the $Ans = 2\pi a$ How to obtain the ans? The arc length formula is used to find the length of an arc of a circle. To find the arc length, we now need to find the circumference It can be understood, that the arc length is a fraction of the circumference of the circle. But it can, at least, be enjoyable. The length of the circumference is given by the formula: C = d, where d is the diameter of the circle. I would like to calculate the arc length of a circle segment, i.e. The length of an arc is a connected section of the circumference of a circle. These curves are called rectifiable and the number is defined as the arc length. Use the formula C = d to calculate the circumference of a circle when the diameter is given. Question: Find the arc length of the circle given by $x^2+y^2=a^2$. Arc length of a circle is the distance measured as the length. For a circle: Arc Length = r (when is in radians) Fun math practice! I have no ideas after doing the following thing. Formula is S = r. See note at end of page. How to Determine the Length of an Arc. I have no ideas after doing the following thing. This formula can also be given as: C = 2r, where r is the radius. Step 2 : Calculation of an arc length without its central angle is a tough problem since the arc length is based on the angle.	{ "domain": "co.uk", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9865717452580314, "lm_q1q2_score": 0.8964167504529535, "lm_q2_score": 0.9086179031191509, "openwebmath_perplexity": 291.45016114516045, "openwebmath_score": 0.90330970287323, "tags": null, "url": "http://stewswebsolutions.co.uk/journal/gz9p0.php?b2be3c=how-to-find-arc-length-of-a-circle" }
Copyright 2017 how to find arc length of a circle	{ "domain": "co.uk", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9865717452580314, "lm_q1q2_score": 0.8964167504529535, "lm_q2_score": 0.9086179031191509, "openwebmath_perplexity": 291.45016114516045, "openwebmath_score": 0.90330970287323, "tags": null, "url": "http://stewswebsolutions.co.uk/journal/gz9p0.php?b2be3c=how-to-find-arc-length-of-a-circle" }
Looking for more algorithms for quasi-random numbers 11-29-2019, 01:06 PM (This post was last modified: 11-30-2019 06:16 AM by Namir.) Post: #1 Namir Senior Member Posts: 690 Joined: Dec 2013 Looking for more algorithms for quasi-random numbers Hi All Math Lovers, In another thread of mine, ttw mentions quasi-random numbers. Quasi-random numbers (QRNs) present a better spread over a range of values than pseudo-random numbers (PRNs). On the other hand, QRNs will often fail randomness tests. They true purpose to to cover more uniformly a range of values in one of more dimensions. This is part of ttw's response in my other thread, where he mentions QRNs: Quote:The easiest multi-dimensional quasi-random sequence is the Richtmeyer sequence. One uses the fractional part of multiples of the square roots of primes. Sqrt(2), Sqrt(3), etc. It's quick to do these by just setting x(i)=0 updating by x(i)=Frac(x(i)+Sqrt(P(i))). Naturally one just stores the fractional parts of the irrationals and updates. (List mode). The sequence is also called the Kronecker or Weyl sequence at times. The above text includes the algorithm of setting x(1)=0 updating by x(i)=Frac(x(i)+Sqrt(P(i))). The array of P() represents prime numbers starting with 2. You can change x(1) to had a uniform random number as a seed (to generate different sequences every time you apply the algorithm) or simply set x(1) = sqrt(P(1)) = sqrt(2). I am curious about other formulas to calculate sequences of quasi-random numbers. You are welcome to use your imagination. My first attempt was something like: Code: n = number of x to generate m = 100n Calculate P() for primes in the range of 1 to m X(1) = rand or Frac(ln(P(3)) sqrt(P(1)) j = 2 count = 0 for i=2 to n X(i) = Frac(X(i) + ln(P(j+1)) * sqrt(P(j-1)) j = j + 1 if j > m then count = count + 1 j = 2 + count end end	{ "domain": "hpmuseum.org", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.986571747626947, "lm_q1q2_score": 0.8964167440608416, "lm_q2_score": 0.9086178944582997, "openwebmath_perplexity": 2756.027968051556, "openwebmath_score": 0.6563873291015625, "tags": null, "url": "https://www.hpmuseum.org/forum/thread-14070-post-124365.html" }
The above code produces x() with a mean near 0.5 and standard deviation near 0.28. The auto correlations for the first 50 lags are in the orde rof 10^(-2) to 10^(-4). I am curious about other formulas to calculate sequences of quasi-random numbers. You are welcome to use your imagination. You can even commit math heresy!!! As long as it works, you are fine (and forgiven) :-) Namir 11-29-2019, 04:49 PM (This post was last modified: 11-29-2019 06:24 PM by SlideRule.) Post: #2 SlideRule Senior Member Posts: 1,013 Joined: Dec 2013 RE: Looking for more algorithms for quasi-random numbers Perusal of Quasi-random sequences in art and integration, John D. Cook Consulting, illumes the phenomena with references to more descriptive books; Random Number Generation and Quasi-Monte Carlo Methods & Monte Carlo and Quasi-Monte Carlo Methods, on the same. BEST! SlideRule 11-30-2019, 01:36 AM Post: #3 mfleming Senior Member Posts: 498 Joined: Jul 2015 RE: Looking for more algorithms for quasi-random numbers (11-29-2019 01:06 PM)Namir Wrote: This is part of ttw's response in my other thread, where he mentions QRNs: Quote:The easiest multi-dimensional quasi-random sequence is the Richtmeyer sequence. One uses the fractional part of multiples of the square roots of primes. Sqrt(2), Sqrt(3), etc. It's quick to do these by just setting x(i)=0 updating by x(i)=Frac(x(i)+Sqrt(P(i))). Naturally one just stores the fractional parts of the irrationals and updates. (List mode). The sequence is also called the Kronecker or Weyl sequence at times. Using "quote" in place of "code" will autowrap large blocks of text! Who decides? 11-30-2019, 01:29 PM Post: #4 Namir Senior Member Posts: 690 Joined: Dec 2013 RE: Looking for more algorithms for quasi-random numbers (11-30-2019 01:36 AM)mfleming Wrote: (11-29-2019 01:06 PM)Namir Wrote: This is part of ttw's response in my other thread, where he mentions QRNs:	{ "domain": "hpmuseum.org", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.986571747626947, "lm_q1q2_score": 0.8964167440608416, "lm_q2_score": 0.9086178944582997, "openwebmath_perplexity": 2756.027968051556, "openwebmath_score": 0.6563873291015625, "tags": null, "url": "https://www.hpmuseum.org/forum/thread-14070-post-124365.html" }
Quote:The easiest multi-dimensional quasi-random sequence is the Richtmeyer sequence. One uses the fractional part of multiples of the square roots of primes. Sqrt(2), Sqrt(3), etc. It's quick to do these by just setting x(i)=0 updating by x(i)=Frac(x(i)+Sqrt(P(i))). Naturally one just stores the fractional parts of the irrationals and updates. (List mode). The sequence is also called the Kronecker or Weyl sequence at times. Using "quote" in place of "code" will autowrap large blocks of text! I learned that the hard way :-) 11-30-2019, 07:52 PM Post: #5 Namir Senior Member Posts: 690 Joined: Dec 2013 RE: Looking for more algorithms for quasi-random numbers The few leads I got from the nice folks on this web were able to lead me to methods that generate sequences of quasi-random numbers that are practically perfectly distributed. I got what I was looking for. Thanks!!! Namir 12-01-2019, 05:52 AM Post: #6 ttw Member Posts: 186 Joined: Jun 2014 RE: Looking for more algorithms for quasi-random numbers This is one of the sequences from my paper in "Computational investigations of low-discrepancy point sets II" from the 1994 Las Vegas Conference on Monte Carlo and Quasi Monte Carlo Methods. I have made a single ad-hoc change (described below) that improves distribution for small numbers of points.	{ "domain": "hpmuseum.org", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.986571747626947, "lm_q1q2_score": 0.8964167440608416, "lm_q2_score": 0.9086178944582997, "openwebmath_perplexity": 2756.027968051556, "openwebmath_score": 0.6563873291015625, "tags": null, "url": "https://www.hpmuseum.org/forum/thread-14070-post-124365.html" }
The Halton Sequence Phi(N,P) (for odd primes, 2 is a special case not considered here) can be described as: 1. Generate the digits of N in base P (for P an odd prime). Call these digits a(1) to a(k) where k is the maximum number of digits needed. (There should be lots of subscripts but I'll treat each prime separately to reduce index management.) N=Sum from j=1 to k of a(j)*P^(j-1), that is: a(k)a(k-1)...a(2)(a(1). 2. Reverse the digits: a(1),a(2)....a(k-1),a(k) is resulting string. 3. Treat this string as a fraction with a decimal point (p-ary point?) in front. Example: P=3, N=5: 5(3)=12. Reverse Phi(5,3)=.21(3) or 2/3+1/9 = 7/9 The sequence is very well distributed for example one starts with 0, 1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27, 10,27... This sequence is uniformly in the unit d-cube using d different primes. For example in 3 dimensions using primes 3 and 5 gives the points: (skipping 0 which sits on the corner of the cube). (1/3, 1/5, 1/7) (2/3, 2/5, 2/7) (1/9, 3/5, 3/7) (4/9, 4/5, 4/7) (7/9,1/25,5/7) (2/9, 6/25, 6/7) (5/9, 11/25, 1/49) (8/9. 16/25, 8/49) etc. The process is sometimes termed a Kakatumi-von Neumann odometer. There is a problem that I noticed about 1967 or so when I started working on quasi-Monte Carlo. For large base, the Halton Sequence produces strongly correlated points until enough points are generated. (This happens with all quasi-random sequences but not as severely.) Take the first few points using bases 101 and 103. (1/101, 1/103) (2/101, 2/103) ... (100/101, 100/103) (1/10201, 101/103) (102/10201, 102/103) (203/10203, 1/10609) etc. In 1993-1995 period, I figured out to multiply each numerator by a number (I called a spin) to break this up. Then I used the fact the fractional parts of square roots of primes are independent to do the following seemingly strange rule.	{ "domain": "hpmuseum.org", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.986571747626947, "lm_q1q2_score": 0.8964167440608416, "lm_q2_score": 0.9086178944582997, "openwebmath_perplexity": 2756.027968051556, "openwebmath_score": 0.6563873291015625, "tags": null, "url": "https://www.hpmuseum.org/forum/thread-14070-post-124365.html" }
Give a prime P, to find a multiplier S, do the following. 1. Compute the nearest integers to the fractional part of the Sqrt(P), call these H and L (high and low, one is above and one below the number). 2. Compute the continued fraction of H/P and L/P; each generates a string of partial quotients. The multiplier S is the one of these satisfying the following: 3. A. Chose the one for which the sum of the partial quotients is smallest. B. If tied, chose the one with the smallest maximum partial quotient. C. If tied, chose whichever H/P or L/P is closest to the fractional part of the square root. D. Ad Hoc Alert: if P=41, use 16. (To avoid 17/41 being close to 12/29. The only such case in all primes less than 2^32) 4. To generate the modified Phi sequence Phi(N,P,S): generate the digits of N base P as above and reverse. Multiply each digit by S modulo P and sum as above. Examples: 3 dimensions: P=3, 5, 7 have S= 2, 2,and 5 respectively. (2/3, 2/5, 5/7) (1/3, 4/5, 3/7) (2/9, 1/5, 1/7) (8/9, 3/4, 6/7) etc. For the pathological case 101 and 103, the multipliers are 6 and 16 respectively (not the best but that's another post sometime). (6/101, 16/103) (12/101, 32/103) (18/101, 48/103) Clearly more spread out than the original Halton Sequence. I've got some more but Mordechay Levin's paper ArXiv 1806 shows that even the original Halton Sequence hits the theoretical lower bound for dimensions 2 and up so great changes cannot be had by tinkering. I do have a bit better, but it's even longer to compute and I haven't tested the new ideas thoroughly. HP50g code:(Number, Prime, Spin, Top, Bottom) << 0 1 -> N P S T B << WHILE N REPEAT N S * P MOD T P * + 'T' STO P 'B' STO* N P IQUOT 'N' STO END T B />> >> Not necessarily the fastest but eas to work with. (If I've done this right.) 12-01-2019, 11:46 AM Post: #7 Csaba Tizedes Senior Member Posts: 412 Joined: May 2014 RE: Looking for more algorithms for quasi-random numbers « Next Oldest \| Next Newest »	{ "domain": "hpmuseum.org", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.986571747626947, "lm_q1q2_score": 0.8964167440608416, "lm_q2_score": 0.9086178944582997, "openwebmath_perplexity": 2756.027968051556, "openwebmath_score": 0.6563873291015625, "tags": null, "url": "https://www.hpmuseum.org/forum/thread-14070-post-124365.html" }
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# Max Sum Of 2 Arrays	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
Reductions. It also prints the location or index at which maximum element occurs in array. int [] A = {−2, 1, −3, 4, −1, 2, 1, −5, 4}; Output: contiguous subarray with the largest sum is 4, −1, 2, 1, with sum 6. Array is an arranged set of values of one-type variables that have a common name. Yes you can find the maximum sum of elements in linear time using single traversal of the array. You can also use the following array formulas: Enter this formula into a blank cell, =SUM(LARGE(A1:D10,{1,2,3})), and then press Ctrl + Shift + Enter keys to get your result. It should have 3 input parameters array A, length and width. Finding the Average value of an Array. Note that in the calculation of max4, we have passed a two dimensional array containing two rows of three elements as if it were a single dimensional array of six elements. In this example, you create two arrays, DAYS and HOURS. Algorithms in Java Assignment: Maximum Sum (in 2 Dimensions) The Problem Given a 2-dimensional array of positive and negative integers, find the sub-rectangle with the largest sum. However, I would like to use the max of these scores. When common element is found then we will add max sum from both the arrays to result. If x and y are scalars and A and B are matrices, y x, A x, and x A have their usual mathematical meanings. Max sum in an array. For all possible combinations, find the sum and compare it with the previous sum and update the maximum sum. Idea is to use merge sort algorithm and maintain two sum for 1st and 2nd array. Search in Rotated Sorted Array. The user will enter a number indicating how many numbers to add and then the user will enter n numbers. I need to check an array of random integers (between 1 and 9) and see if any combination of them will add up to 10. A corner element is an element from the start of the array or from the end of the array. Once the type of a variable is declared, it can only store a value belonging to this particular type. Create a max heap i.	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
is declared, it can only store a value belonging to this particular type. Create a max heap i. Given an array, you have to find the max possible two equal sum, you can exclude elements. Question E3: WAP to find out the row sum and column sum of a two dimensional array of integers. Whenever possible, make sure that you are using the NumPy version of these aggregates when operating on NumPy arrays!. Write a program to find those pair of elements that has the maximum and minimum difference among all element pairs. Input size and elements in array, store in some variable say n and arr[n]. Here is the complete Java program with sample outputs. Algorithms in Java Assignment: Maximum Sum (in 2 Dimensions) The Problem Given a 2-dimensional array of positive and negative integers, find the sub-rectangle with the largest sum. Method since it requires contiguous, it means that for each element, it has two situations that are in the subarray or not. Easy Tutor says. The return value of min () and max () functions is based on the axis specified. C++ :: Creating Table Of Arrays - Find Maximum Value And Sum Aug 12, 2014. Google Advertisements. Sample Run: [2, 1, 8, 4, 4] Min: 1 Max: 8 Average: 3. The master will loop from 2 to the maximum value on issue MPI_Recv and wait for a message from any slave (MPI_ANY_SOURCE), if the message is zero, the process is just starting, if the message is negative, it is a non-prime, if the message is positive, it is a prime. Given an integer array of N elements, find the maximum sum contiguous subarray (containing at least one element). HackerRank Solutions Over the course of the next few (actually many) days, I will be posting the solutions to previous Hacker Rank challenges. 1 Answer to Given that A[MAX_ROWS][MAX_COLUMNS] is a 2 dimensional array of integers write a C ++ function. WriteLine to do this. SUMPRODUCT( array1, [array2, array_n] ) Parameters or Arguments array1, array2, array_n The ranges of cells or arrays that you wish to multiply.	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
or Arguments array1, array2, array_n The ranges of cells or arrays that you wish to multiply. A better solution would be to find the two largest elements in the array, since adding those obviously gives the largest sum. The maximum product is formed by the (-10, -3) or (5, 6) pair. Algorithms in Java Assignment: Maximum Sum (in 2 Dimensions) The Problem Given a 2-dimensional array of positive and negative integers, find the sub-rectangle with the largest sum. A selected portion of the array may be summed, if an integer range expression is provided with the array name (. max (x) → [same as input] Returns the maximum value of all input values. Write a program to find sum of each digit in the given number using recursion. Basic Operations ¶. Find the sum of numbers and represent it in array. The function should return an integer. A one-dimensional array is like a list; A two dimensional array is like a table; The C language places no limits on the number of dimensions in an array, though specific implementations may. Write a program to find top two maximum numbers in a array. MS Excel 2007: Use an array formula to sum all of the order values for a given client This Excel tutorial explains how to use an array formula to sum all of the order values for a given client in Excel 2007 (with screenshots and step-by-step instructions). min () find the maximum and minimum value of the arguments, respectively. This very simply starts with a sum of 0 and add each item in the array as we go: public static int findSumWithoutUsingStream (int[] array) { for (int value : array) { 2. In this article we’ll explore four plug and play functions that allow you to easily find certain values in an arrays of numbers. (For clarification, the L-length subarray could occur before or after the M-length subarray. A matrix with m rows and n columns is actually an array of length m, each entry of which is an array of length n. We can switch from one array to another array only at common elements.	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
is an array of length n. We can switch from one array to another array only at common elements. For example if input integer array is {2, 6, 3, 9, 11} and given sum is 9, output should be {6,3}. See (2) in the diagram. We can start from either arrays but we can switch between arrays only through its common elements. My solution for the bigDiff using the inbuilt Math. If you sum the second array you can use that to multiply the first array because that will be the same as multiplying the values individually and then summing the results. K maximum sum combinations from two arrays Given two equally sized arrays (A, B) and N (size of both arrays). Objective Problem Statement • Application of parallel prefix: Identifying the maximum sum that can be computed using. min (x) → [same as input]. Pop the heap to get the current largest sum and along. Maximum Sum of Two Non-Overlapping Subarrays. For example, to sum the top 20 values in a range, a formula must contain a list of integers from 1 to 20. For example [1,3,5,6,7,8,] here 1, 3 are adjacent and 6, 8 are not adjacent. We are making max_sum_subarray is a function to calculate the maximum sum of the subarray in an array. Specifically we'll explore the following: Finding the Minimum value in an Array. 4+ PHP Changelog: PHP versions prior to 4. And so myself and the OP exchanged a comment: I have concern. For an array x, y=cumsum(x) returns in the scalar y the cumulative sum of all the elements of x. In C programming, you can pass en entire array to functions. Input the array elements. You may have A1:A20, then A30:A35 filled. Thus, two arrays are “equal” according to Array#<=> if, and only if, they have the same length and the value of each element is equal to the value of the corresponding element in the other array. C++ Programs to Delete Array Element C++ Programs to Sum of Array Elements. Find the sum of the maximum sum path to reach from beginning of any array to end of any of the two arrays. Top Forums Shell	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
sum path to reach from beginning of any array to end of any of the two arrays. Top Forums Shell Programming and Scripting Sum elements of 2 arrays excluding labels Post 303015114 by Don Cragun on Wednesday 28th of March 2018 06:34:58 AM. An index value of a Java two dimensional array starts at 0 and ends at n-1 where n is the size of a row or column. Given input array be,. In this solution dp* stores the maximum among all the sum of all the sub arrays ending at index i. Once the type of a variable is declared, it can only store a value belonging to this particular type. log10(a) Logarithm, base 10. This function subtracts when negative numbers are used in the arguments. (2-D maximum-sum subarray) (30 points) In the 2-D Maximum-Sum Subarray Prob- lem, you are given a two-dimensional m x n array A[1 : m,1: n of positive and negative numbers, and you are asked to find a subarray Ala b,c: 1 Show transcribed image text Expert Answer. C Program to read an array of 10 integer and find sum of all even numbers. if 2,3,4, 5 is the given array, {4,5,2,3} is also a possible array like other two. Our maximum subset sum is. SemanticSpace Technologies Ltd interview question: There is an integer array consisting positive numbers only. If any element is greater than max you replace max with. Enables ragged arrays. Array-2, Part I ”. I haven't gotten that far yet, I'm stuck just trying to print my two arrays, every time i try to print the first array it gives me the elements of the second array and it. All arrays must have the same number of rows and columns. i* n/2 – Or overlaps both halfs: i* n/2 j* • We can compute the best subarray of the first two types with recursive calls on the left and right half. Then we compare it with the other array elements one by one, if any element is greater than our assumed. I need to check an array of random integers (between 1 and 9) and see if any combination of them will add up to 10. The maximum product is formed by the (-10, -3) or (5, 6)	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
any combination of them will add up to 10. The maximum product is formed by the (-10, -3) or (5, 6) pair. SUM (C, DIM=1) returns the value (5, 7, 9), which is the sum of all elements in each column. computes bitwise conjunction of the two arrays (dst = src1 & src2) Calculates the per-element bit-wise conjunction of two arrays or. This program shows you how to find the sum of rows and columns in a two dimensional array, and also how to use a method to calculate the sum of every element inside of a 2d array. Given two equally sized arrays (A, B) and N (size of both arrays). We can do this by using or without using an array. (Array): Returns the new array of chunks. If they are even we will try to find whether that half of sum is possible by adding numbers from the array. Latest commit message. So, the minimum of the array is -1 and the maximum of the array is 35. Google Advertisements. 999997678497 499911. Add solution to Pairs problem. Next, we use a standard for loop to iterate through our array numbers and total up the. Binary Tree Maximum Path Sum Lowest Common Ancestor I II III Binary Tree Level Order Traversal I II Kth Smallest Sum In Two Sorted Arrays LinkedList. We can switch from one array to another array only at common elements. Bottleneck code often involves condi-tional logic. if the sum of previous subarray is negative, it means that it need. 1<=Ai<=10000, where Ai is the ith integer in the array. The purpose of the SUMPRODUCT function is to multiply, then sum, arrays. The page is a good start for people to solve these problems as the time constraints are rather forgiving. Also add the common element to the result. The algorithm to find maximum is: we assume that it's present at the beginning of the array and stores that value in a variable. It may or may not include a[i-1], a[i-2], etc. M = max( A ,[], 'all' , nanflag ) computes the maximum over all elements of A when using the nanflag option. We can update both incrementally by counting from the back,	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
of A when using the nanflag option. We can update both incrementally by counting from the back, so we have to keep track of two things: \$\max(S[i:])\$ and \$\max(B[i+1:])\$. Specifically we’ll explore the following: Finding the Minimum value in an Array. We take a two dimensional array L of size count+1, sum/2+1. Partition an array into two sub-arrays with the same sum. For example, given array A such that: A[0] = 3 A[1] = 2 A[2] = -6 A[3] = 4 A[4] = 0. Each element, therefore, must be accessed by a corresponding number of index values. creating a recursive way to find max and min in array I am kind of confused with this instruction: Describe a recursive algorithm for finding both the minimum and maximum elements in an array A of n elements. If ARRAY is a zero-sized array, the result equals zero. computes the sum of two matrices and then prints it. The basis is p[0] = a[0]. package net. What is the sum of the last column? 5. Yes you can find the maximum sum of elements in linear time using single traversal of the array. In order to find the sum of all elements in an array, we can simply iterate the array and add each element to a sum accumulating variable. The first thing that we tend to need to do is to scan through an array and examine values. sum(my_first_array) >my_first_array. Our maximum subset sum is. Here is the complete Java program with sample outputs. Naive solution would be to consider every pair of elements and calculate their product. the contiguous subarray [4,-1,2,1] has the largest sum = 6. Question E2: WAP to display the values of a two dimensional array in the matrix form. Suppose we need to find out max sum subarray. As a "rule of thumb", any "calculated array" - in this case the array calculated by adding two ranges - results in a formula that requires CSE, although some functions (like SUMPRODUCT and LOOKUP) don't normally need CSE even with calculated arrays - to allow normal entry you can add an INDEX function - I edited my answer to the	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
calculated arrays - to allow normal entry you can add an INDEX function - I edited my answer to the effect. maxSubsetSum has the following parameter(s): arr: an array of integers. amax() by thispointer. Given an array A of non-negative integers, return the maximum sum of elements in two non-overlapping (contiguous) subarrays, which have lengths L and M. Solution to Question 2. We can use an array as a deque with the following operations:. Improvement over Method-1 – O(n 2) Time. If only ARRAY is specified, the result equals the sum of all the array elements of ARRAY. It works as follows. Once the type of a variable is declared, it can only store a value belonging to this particular type. Maximize array sum by concatenating corresponding elements of given two arrays Given two array A[] and B[] of the same length, the task is to find the maximum array sum that can be formed by joining the corresponding elements of the array in any order. Our maximum subset sum is. In a two-dimensional Java array, we can use the code a[i] to refer to the ith row (which is a one-dimensional array). And so myself and the OP exchanged a comment: I have concern. I borrowed some code from other forums that had similar programs, but obviously it doesn't match my needs specifically. Given an array of integers, find maximum product of two integers in an array. Computes the matrix multiplication of two arrays. Add solutions to C++ domain. This problem is generally known as the maximum sum contiguous subsequence problem and if you haven’t encountered it before, I’d recommend trying to solve it before reading on. #include using namespace std; int main() { const int SIZE = 12; double months[SIZE]; int count; double sum = 0; double totalRainfall; double. c++: which functions gives the sum of an array? (3). In this example, we will find the sum of all elements in a numpy array, and with the default optional parameters to the sum () function. min (x) → [same as input]. Given two arrays of positive	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
optional parameters to the sum () function. min (x) → [same as input]. Given two arrays of positive integers. For example, for the sequence of values −2, 1, −3, 4, −1, 2, 1, −5, 4; the contiguous subarray with the largest sum is 4, −1, 2, 1, with sum 6. Let arr[i. Your code tries all \$n (n+1)/2 \$ combinations of array elements to find the combination with the largest sum, so the complexity is \$O(n^2) \$. The loop structure should look like for (i=0; i=2 and find the sum of smallest and second smallest, then our answer will be maximum sum among them. Find the sum of the maximum sum path to reach from beginning of any array to end of any of the two arrays. Maximum Sum of Two Non-Overlapping Subarrays 2019/04/22 2019/04/22 shiji Leetcode Given an array A of non-negative integers, return the maximum sum of elements in two non-overlapping (contiguous) subarrays, which have lengths L and M. Write a function to find the maximum sum of all subarrays. In order to find the sum of all elements in an array, we can simply iterate the array and add each element to a sum accumulating variable. If all the array entries were positive, then the maximum-subarray problem would present no challenge, since the entire array would give the greatest sum. The length property is the array length or, to be precise, its last numeric index plus one. I borrowed some code from other forums that had similar programs, but obviously it doesn't match my needs specifically. The maximum product is formed by the (-10, -3) or (5, 6) pair. 7 is the sum of 2 + 5 in column 2, and so forth. Sub Array with Maximum Sum – Kadane Algorithm is the best solution. Note : Imp to execute and trace to understand and remember. Edit: given your comments if the initial array is fixed then you can use MMULT function like this. Passing array elements to a function is similar to passing variables to a function. if orientation is equal to n then. M=sum(A,dim) In Scilab dim=1 is equivalent to dim="r" and dim=2 is equivalent	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
is equal to n then. M=sum(A,dim) In Scilab dim=1 is equivalent to dim="r" and dim=2 is equivalent dim="c". For examples, Enter 1st integer: 8 Enter 2nd integer: 2 Enter 3rd integer: 9 The sum is: 19 The product is: 144 The min is: 2 The max is: 9 Hints. Read the entered array size and store that value into the variable n. Maximum Sum Subarray (In Yellow) For example, for the array given above, the contiguous subarray with the largest sum is [4, -1, 2, 1], with sum 6. How to swap two numbers without using temporary variable? Write a program to print fibonacci series. Find maximum possible sum of elements such that there are no 2 consecutive elements present in the sum. Find ways to calculate a target from elements of specified. Given an array, find maximum sum of smallest and second smallest elements chosen from all possible sub-arrays. Once you have a vector (or a list of numbers) in memory most basic operations are available. Find the sum of the maximum sum path to reach from beginning of any array to end of any of the two arrays. So, the minimum of the array is -1 and the maximum of the array is 35. The maximum admissible amount of dimensions in an array is four. For example, A = [−2, 1, −3, 4, −1, 2, 1, −5, 4] then max sum=11 with the subarray [1, 4, 2, 4]. Calculates the per-element sum of two arrays or an array and a scalar. In order to find the sum of all elements in an array, we can simply iterate the array and add each element to a sum accumulating variable. Join 124,729,115 Academics and Researchers. Expected time complexity is O(m+n) where m is the number of elements in ar1[] and n is the number of elements in ar2[]. I am trying to compute the maximum possible sum of values from a matrix or 2d array or table or any suitable structure. A selected portion of the array may be summed, if an integer range expression is provided with the array name (. Hence there would be four different arrays in this case. sum += numbers [i] In The Standard Way we first declare	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
would be four different arrays in this case. sum += numbers [i] In The Standard Way we first declare the variable sum and set its initial value of zero. The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f. Given an array A of non-negative integers, return the maximum sum of elements in two non-overlapping (contiguous) subarrays, which have lengths L and M. SUM (C, DIM=2) returns the value (6, 15), which is the sum of all elements in each row. e 1,2,3,4,6 is given array we can have max two equal sum as 6+2 = 4+3+1. =MAX(IF((List>=LLim)*(List<=ULim),List,FALSE)) returns the maximum of values between 2 and 5, or 5. When you want to return a sum for a single criteria (for example, a single IF condition) When you want to use multiple criteria and return the sum to multiple cells; The criteria that you can use with the SUMIF() worksheet function is limited to text, numbers, or a range, and the function cannot use array constants. Stack PUSH & POP Implementation using Arrays; Program to remove duplicate element in an array; C Program to sort the matrix rows and columns; Write a c program for swapping of two arrays; C Program to read name and marks of students and store it in file; To find out the maximum number in an array using function. Explore Channels Plugins & Tools Pro Login About Us. Academia is the easiest way to share papers with millions of people across the world for free. The min () and max () functions of numpy. A corner element is an element from the start of the array or from the end of the array. Search in Rotated Sorted Array II. I need to create a table of this which i have done using case 1. The basis is p[0] = a[0]. (For clarification, the L-length subarray could occur before or after the M-length subarray. For example, consider the array {-10, -3, 5, 6, -2}. Calculates the per-element sum of two arrays or an array and a scalar. Display the maximum K valid sum combinations from all the possible sum	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
or an array and a scalar. Display the maximum K valid sum combinations from all the possible sum combinations. Here we are setting up the pointer to the base address of array and then we are incrementing pointer and using * operator to get & sum-up the values of all the array elements. Even if you have encountered it before, I’ll invite you. For example, to sum the top 20 values in a range, a formula must contain a list of integers from 1 to 20. If the current element of array 1 and array 2are same, then take the maximum of sum1 and sum2 and add it to the result. In the Java programming language, a multidimensional array is an array whose components are themselves arrays. For example, entering =SUM(10, 2) returns 12. How to swap two numbers without using temporary variable? Write a program to print fibonacci series. Yes you can find the maximum sum of elements in linear time using single traversal of the array. To store sum of array elements, initialize a variable sum = 0. Latest commit 7b136cc on Mar 10, 2019. The function should return an integer. The call to new Array(number) creates an array with the given length, but without elements. What's wrong with the scrap of code in the question? The array is of size 5, but the loop is from 1 to 5, so an attempt will be made to access the nonexistent element a[5]. def array_summer(arr): return sum (arr) # Test input print (array_summer ( [1, 2, 3, 3, 7])) we went through two different methods of summing the elements of an array. For an array x, y=cumsum(x) returns in the scalar y the cumulative sum of all the elements of x. computes the sum of two matrices and then prints it. C++ :: Creating Table Of Arrays - Find Maximum Value And Sum Aug 12, 2014. Basicly I have to sum rows and columns of 2d array and then store the results in separate arrays, as far as the code is now, can see it quite clearly. Finding the Maximum value in an Array. ; The array formula lets the IF function test for multiple conditions in a single	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
value in an Array. ; The array formula lets the IF function test for multiple conditions in a single cell, and, when the data meets a condition, the array formula determines what data (event results) the MAX function will examine to find the best result. Yes you can find the maximum sum of elements in linear time using single traversal of the array. The previous contiguous array sum was less than or equals 0. Finally, if A is a multidimensional array, Matlab works on the first non-singleton dimension of A what Scilab does not. with - sum of two arrays in c. With the following program, you can even print the sum of two numbers or three numbers up to N numbers. For examples, Enter 1st integer: 8 Enter 2nd integer: 2 Enter 3rd integer: 9 The sum is: 19 The product is: 144 The min is: 2 The max is: 9 Hints. Finding the Sum of all values in an Array. For example [1,3,5,6,7,8,] here 1, 3 are adjacent and 6, 8 are not adjacent. For an array x, y=cumsum(x) returns in the scalar y the cumulative sum of all the elements of x. Note : Imp to execute and trace to understand and remember. The problem: given an array which could contain zero, negative, and positive numbers, find the largest sum of contiguous sub-array. Move the pointer in the corresponding heap there. We can update both incrementally by counting from the back, so we have to keep track of two things: \$\max(S[i:])\$ and \$\max(B[i+1:])\$. Array-2, Part I ”. To get the sum of all elements in a numpy array, you can use Numpy’s built-in function sum (). Question 3. such that sum(wi*xi)<=W & x(0,1) The unbounded knapsack problem (UKP) places no upper bound on the number of copies of each kind of item. Maximum Sum Subarray (In Yellow) For example, for the array given above, the contiguous subarray with the largest sum is [4, -1, 2, 1], with sum 6. These functions will not work as-is with arrays of numbers. computes the sum of two matrices and then prints it. Given an array A of non-negative integers, return the maximum	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
of two matrices and then prints it. Given an array A of non-negative integers, return the maximum sum of elements in two non-overlapping (contiguous) subarrays, which have lengths L and M. Add solutions to C domain. Two Dimensional Array in C Example. Dim i,sum, r As Integer sum=0 For i=0 to 4 ‘assign values to the array x(i)=i*i Next r=1 For each v in x ‘read the elements of the array in to Excel cells MsgBox v r= r+1 Next Note: the start index of the array is 0 and its size is equal to last index added by 1. Once the type of a variable is declared, it can only store a value belonging to this particular type. Array formulas can also be used find out the maximum and minimum values for a given set of conditions. The call to new Array(number) creates an array with the given length, but without elements. Maximum of array elements over a given axis. Sub Array with Maximum Sum – Kadane Algorithm is the best solution. The SUM function in Excel adds the arguments you enter in a formula. Java Program to Find The Sum of Array Elements \|\| D. sum() 15 How to find the maximum value in NumPy 1d-array? We can find the maximum value stored in. Subarr2[] = {3, 3, 12. Problem Statement: Given an array A = {a 1, a 2, a 3, , a N} of N elements, find the maximum possible sum of a. As a "rule of thumb", any "calculated array" - in this case the array calculated by adding two ranges - results in a formula that requires CSE, although some functions (like SUMPRODUCT and LOOKUP) don't normally need CSE even with calculated arrays - to allow normal entry you can add an INDEX function - I edited my answer to the effect. Print the N integers of the array in the reverse order in a single line separated by a space. Which row has the largest sum? 4. Array is an arranged set of values of one-type variables that have a common name. Take a HashMap with Key and value as Integer types. This problem is generally known as the maximum sum contiguous subsequence problem and if you haven’t encountered it	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
generally known as the maximum sum contiguous subsequence problem and if you haven’t encountered it before, I’d recommend trying to solve it before reading on. The sum of all the numbers in the array. To store sum of array elements, initialize a variable sum = 0. C++ Programs to Delete Array Element C++ Programs to Sum of Array Elements. Let arr[i. Where L[i,j]=maximum sum possible with elements of array 0 to i and sum not exceeding j. Using only one loop, Complete the code to compute both sums. Function Description. Description: ----- If we add amount of max INT with number 1 in array_sum function , the result will be false. Hi, My documents have an "aliases" field which is an array of string. Objective: The maximum subarray problem is the task of finding the contiguous subarray within a one-dimensional array of numbers which has the largest sum. A better solution would be to find the two largest elements in the array, since adding those obviously gives the largest sum. Find Maximum sum sub array of tempArray. I'm stumped. if the array was [5, 6, 4, 2, 9] it would return true. j* n/2 – Or contained entirely in the right half , i. For all possible combinations, find the sum and compare it with the previous sum and update the maximum sum. In this example, we will find the sum of all elements in a numpy array, and with the default optional parameters to the sum () function. To find the maximum value, you initialize a placeholder called max with the value of the first element in the array. 1 Answer to Given that A[MAX_ROWS][MAX_COLUMNS] is a 2 dimensional array of integers write a C ++ function. int a[3]; // creates an array with 'Numb' elements a[3] = 5; // assigns 5 to index 3 (the 4th element) in the array This is effectively what you're doing with your cin line. But when you try to follow the same approach with an array formula, Excel complains. Problem Description We have to write a program in C such that the program will allocate 2 one-dimensional arrays using	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
We have to write a program in C such that the program will allocate 2 one-dimensional arrays using malloc() call and then will do the addition and stores the result into 3rd array. min(), big_array. but it must include a[i]. Maximum sum in circular array such that no two elements are adjacent \| Set 2 Given an array arr[] of positive numbers, find the maximum sum of a subsequence with the constraint that no 2 numbers in the sequence should be adjacent in the array where the last and the first elements are assumed adjacent. Latest commit 7b136cc on Mar 10, 2019. How we can do that efficiently?. max (x, n) → array<[same as x]> Returns n largest values of all input values of x. The elements entered in the array are as follows: 1 2 35 0 -1. The prior values added up to 0, meaning that the new max subarray starts from this value. Empty subarrays/subsequences should not be considered. See (2) in the diagram. For examples, Enter 1st integer: 8 Enter 2nd integer: 2 Enter 3rd integer: 9 The sum is: 19 The product is: 144 The min is: 2 The max is: 9 Hints. For example if input integer array is {2, 6, 3, 9, 11} and given sum is 9, output should be {6,3}. (#M40034130) C Programming question Find out maximim sum of sub Array Keep an EYE Find out maximim sum of sub Array example array={2,3,-1,4,9} maximum sum of sub array=17 Asked In C DHIRENDRA (6 years ago) Unsolved Read Solution (3) Is this Puzzle helpful? (1) (0) Submit Your Solution Program. The expression within the optional "with" clause can be used to specify the item to use in the reduction. Dynamic Memory Allocation Example: In this C program, we will declare memory for array elements (limit will be at run time) using malloc(), read element and print the sum of all elements along with the entered elements. Return the maximum of sum1 and sum2. 2 Vectorized Logic The previous section shows how to vectorize pure computation. The "waterdrop" camera array is made up of a 48-MP main unit, an 8-MP wide-angle shooter, and a 5-MP	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
The "waterdrop" camera array is made up of a 48-MP main unit, an 8-MP wide-angle shooter, and a 5-MP depth sensor, with a flash module at the bottom (2020) vs iPhone 11, 11 Pro and Pro Max. Full Discussion: How do I find the sum of values from two arrays? Top Forums Shell Programming and Scripting How do I find the sum of values from two arrays? Post 302579313 by kshji on Monday 5th of December 2011 12:03:10 PM. Given two equally sized arrays (A, B) and N (size of both arrays). Algorithms in Java Assignment: Maximum Sum (in 2 Dimensions) The Problem Given a 2-dimensional array of positive and negative integers, find the sub-rectangle with the largest sum. Whenever possible, make sure that you are using the NumPy version of these aggregates when operating on NumPy arrays!. The min () and max () functions of numpy. Pop the heap to get the current largest sum and along. Your code tries all \$n (n+1)/2 \$ combinations of array elements to find the combination with the largest sum, so the complexity is \$O(n^2) \$. Arr2[] = {1, 3, 3, 12, 2} then maximum result is obtained when we create following two subarrays − Subarr1[] = {2, 4, 3} and. Using only one loop, Complete the code to compute both sums. Timing Belt - $1,299. Given an integer array of N elements, find the maximum sum contiguous subarray (containing at least one element). Compute sum of two digit arrays. For example, if A is a matrix, then max(A,[],[1 2]) computes the maximum over all elements in A, since every element of a matrix is contained in the array slice defined by dimensions 1 and 2. here maximum subarray is [2,3,4]. Divide Two Integers 4. You can max_heap both arrays, and set an iterator at the root for both. Complexity Analysis:. Input: nums = [1,1,1], k = 2 Output: 2. If only ARRAY is specified, the result equals the sum of all the array elements of ARRAY. Array exponentiation is available with A. This program is an example of Dynamic Memory Allocation, here we are declaring memory for N array	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
A. This program is an example of Dynamic Memory Allocation, here we are declaring memory for N array elements at run time using malloc() - which is used to declare memory for N. Problem Description Write a program to find the sum of the corresponding elements in 2 arrays. A matrix with m rows and n columns is actually an array of length m, each entry of which is an array of length n. My solution for the bigDiff using the inbuilt Math. def sum_odd(n): if n < 2: return 1 elif n%2 == 0: return sum_odd(n-1) else: return n + sum_odd(n-2) Note that this function returns 1 if n is not greater 0 as is defined in the original function. You can refer to more than one array in a single SAS statement. Given input array be,. Note: Values of different types will be compared using the standard comparison rules. To display sub array with maximum sum you should write code to hold the start and end value of the sub array with maximum sum. My operation system is 64 bit. In this tutorial, I am going to discuss a very famous interview problem find maximum subarray sum (Kadane’s algorithm). The optimal strategy is to pick the elements form the array is, two. Pure VPN Privide Lowest Price VPN Just @$1. Find ways to calculate a target from elements of specified. The first user will be asked to enter the order of the matrix (such as the numbers of rows and columns) and then enter the elements of the two matrices. I am trying to compute the maximum possible sum of values from a matrix or 2d array or table or any suitable structure. In a two-dimensional Java array, we can use the code a[i] to refer to the ith row (which is a one-dimensional array). MS Excel 2007: Use an array formula to sum all of the order values for a given client This Excel tutorial explains how to use an array formula to sum all of the order values for a given client in Excel 2007 (with screenshots and step-by-step instructions). Consider an integer array, the number of elements in which is determined by the user. Find a	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
Consider an integer array, the number of elements in which is determined by the user. Find a Triplet having Maximum Product in an Array. Sum the largest 3 numbers: =SUM(LARGE(range, {1,2,3})) Sum the smallest 3 numbers: =SUM(SMALL(range, {1,2,3})) Don't forget to press Ctrl + Shift + Enter since you are entering the Excel array formula, and you will get the following result: In a similar fashion, you can calculate the average of N smallest or largest values in a range:. Arrays in formulas. Array Maximum Minimum value We can calculate maximum value among the elements of an array by using max function. For example, consider the array {-10, -3, 5, 6, -2}. We can switch from one array to another array only at common elements. Hi, My documents have an "aliases" field which is an array of string. Basic array operations. Space Complexity: O(1). If it's provided then it will return for array of max values along the axis i. A blog about Java, Spring, Hibernate, Programming, Algorithms, Data Structure, SQL, Linux, Database, JavaScript, and my personal experience. Problem Statement: Given an array A = {a 1, a 2, a 3, , a N} of N elements, find the maximum possible sum of a. Add solution to Super Maximum Cost Queries problem. Naive solution would be to consider every pair of elements and calculate their product. Maximize array sum by concatenating corresponding elements of given two arrays Given two array A[] and B[] of the same length, the task is to find the maximum array sum that can be formed by joining the corresponding elements of the array in any order. I have an array of "2,3,4,5,6,9,10,11,12,99". The length property is the array length or, to be precise, its last numeric index plus one. Now the above Leetcode challenge is a special case of the general Max Subarray classic problem in computer science - which is the task of finding the contiguous subarray within a one-dimensional array of numbers which has the largest sum. We will implement a simple algorithm in	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
one-dimensional array of numbers which has the largest sum. We will implement a simple algorithm in javascript to find the maximum sum of products of given two arrays. Yesterday we got the question Sum of Maximum GCD from two arrays. Note that the common elements do not have to be at same indexes. How to swap two numbers without using temporary variable? Write a program to print fibonacci series. Non-Numeric or Non-Existent Fields¶. If DIM is absent, a scalar with the sum of all elements in ARRAY is returned. It loops over the values and returns the sum of the elements. Note that in the calculation of max4, we have passed a two dimensional array containing two rows of three elements as if it were a single dimensional array of six elements. Yes you can find the maximum sum of elements in linear time using single traversal of the array. Jerico January 10, 2014 at 6:30 am. For example, if the array contains: 31, -41, 59, 26, -53, 58, 97, -93, -23, 84 then the largest sum is 187 taken from the [59. The bottommost cell is A35. So far so good, and it looks as if using a list is as easy as using an array. As an example, the maximum sum contiguous subsequence of 0, -1, 2, -1, 3, -1, 0 would be 4 (= 2 + -1 + 3). 12) instead of the number entered into the array. In order to find the sum of all elements in an array, we can simply iterate the array and add each element to a sum accumulating variable. Hence there would be four different arrays in this case. Find maximum sum path involving elements of given arrays Given two sorted array of integers, find a maximum sum path involving elements of both arrays whose sum is maximum. HI everyone, need help with this exercise: "We have two integer numbers, which are represented by two arrays. For all possible combinations, find the sum and compare it with the previous sum and update the maximum sum. Sort both arrays array A and array B. 404 24 Add to List Share. This is way faster than a manually using a for loop going through all	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
B. 404 24 Add to List Share. This is way faster than a manually using a for loop going through all elements in a 1d-array. Monotonic Queue/Stack. Also add the common element to the result. Passing array elements to a function is similar to passing variables to a function. If the current element of array 1 and array 2are same, then take the maximum of sum1 and sum2 and add it to the result. We are making max_sum_subarray is a function to calculate the maximum sum of the subarray in an array. This program will help to understand the working of for loop, array, if statement and random numbers. C program to find the maximum or the largest element and the location (index) at which it's present in an array. Then print the respective minimum and maximum values as a single line of two space-separated long integers. Which is run a loop from 0 to n. If arr1[] = {1, 2, 4, 3, 2} and. If only ARRAY is specified, the result equals the sum of all the array elements of ARRAY. =MAX(IF((List>=LLim)*(List<=ULim),List,FALSE)) returns the maximum of values between 2 and 5, or 5. A better solution would be to find the two largest elements in the array, since adding those obviously gives the largest sum. You can also declare an array of arrays (also known as a multidimensional array) by using two or more sets of brackets, such as String[][] names. To initialize array use random numbers. Input the array elements. Introduction to C Programming Arrays Overview. array_sum: Array Sum of. hence maximum subarray sum is 9. To display sub array with maximum sum you should write code to hold the start and end value of the sub array with maximum sum. if 2,3,4, 5 is the given array, {4,5,2,3} is also a possible array like other two. C++ :: Creating Table Of Arrays - Find Maximum Value And Sum Aug 12, 2014. First Bad Version. Even to find the total number of even elements in the array. This program is an example of Dynamic Memory Allocation, here we are declaring memory for N array elements at run	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
is an example of Dynamic Memory Allocation, here we are declaring memory for N array elements at run time using malloc() - which is used to declare memory for N. When you want to return a sum for a single criteria (for example, a single IF condition) When you want to use multiple criteria and return the sum to multiple cells; The criteria that you can use with the SUMIF() worksheet function is limited to text, numbers, or a range, and the function cannot use array constants. Add solutions to C++ domain. Note that the common elements do not have to be at same indexes. 7 is the sum of 2 + 5 in column 2, and so forth. Java program to calculate the sum of N numbers using arrays, recursion, static method, using while loop. (#M40034130) C Programming question Find out maximim sum of sub Array Keep an EYE Find out maximim sum of sub Array example array={2,3,-1,4,9} maximum sum of sub array=17 Asked In C DHIRENDRA (6 years ago) Unsolved Read Solution (3) Is this Puzzle helpful? (1) (0) Submit Your Solution Program. The code below will show how to display a maximum of 3 items from an integer array. sum = sum + (value at 2000) In the Second iteration we will have following calculation – sum = sum + (value at 2002) = 11 + 12 = 23. We can switch from one array to another array only at common elements. Naive solution would be to consider every pair of elements and calculate their product. If the first and only parameter is an array, max() returns the highest value in that array. After partitioning, each subarray has their values changed to become the maximum value of that subarray. This function can take two other optional arguments, that will be covered in more detail, when we get to multi-dimensional arrays. It is For Each Loop or enhanced for loop introduced in java 1. Here is a simple example. minimum difference = second lowest - lowest. sum () is shown below. The master will loop from 2 to the maximum value on issue MPI_Recv and wait for a message from any slave	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
master will loop from 2 to the maximum value on issue MPI_Recv and wait for a message from any slave (MPI_ANY_SOURCE), if the message is zero, the process is just starting, if the message is negative, it is a non-prime, if the message is positive, it is a prime. Here we will be displaying the sub array. Is there. Given an array, find maximum sum of smallest and second smallest elements chosen from all possible sub-arrays. A better solution would be to find the two largest elements in the array, since adding those obviously gives the largest sum. For example, entering =SUM(10, 2) returns 12. 1- creat two int array the size of each array must be 40 2- ask the user to input 2 strings (big numbers ) 3- check each character of ths string if it's numeric characters 4- add each string in one int array 5-print out the sum of the 2 arrays 6-compare between the two arrays. the contiguous subarray [4,-1,2,1] has the largest sum = 6. c++: which functions gives the sum of an array? (3). if the array was [5, 6, 4, 2, 9] it would return true. Bilal-March 5th, 2020 at 2:07 pm none Comment author #29091 on Find max value & its index in Numpy Array \| numpy. So for the test arrays: int[] testArrayA = { 7, 1, 4, 5, 1}; int[] testArrayB = { 3, 2, 1, 5, 3}; ↑ starting i. Given two sorted arrays such the arrays may have some common elements. Adjacent: side by side. You need to create three different functions called minNumber,maxNumber and totalSum. 4 Two-dimensional Arrays. three two one one three two two three one three one two Array Reduction Methods : Array reduction methods can be applied to any unpacked array to reduce the array to a single value. Compute the ceiling power of 2. maximum difference = higest-lowest. Join 124,729,115 Academics and Researchers. While if we add this two via plus (+) operator ,the result will be true. SUM (C, DIM=2) returns the value (6, 15), which is the sum of all elements in each row. If it's provided then it will return for array of max values along	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
sum of all elements in each row. If it's provided then it will return for array of max values along the axis i. It should return an integer representing the maximum subset sum for the given array. sum %2 [1] 1 1 [2] 2 0 [3] 3 1 [1,2] 3 1 [2,3] 5 1 [1,2,3] 6 0. The syntax of numpy. max (x, n) → array<[same as x]> Returns n largest values of all input values of x. For examples, Enter 1st integer: 8 Enter 2nd integer: 2 Enter 3rd integer: 9 The sum is: 19 The product is: 144 The min is: 2 The max is: 9 Hints. Which is run a loop from 0 to n. Search in Rotated Sorted Array. Sort both arrays array A and array B. The bottommost cell is A35. 15 is the sum of 4 + 5 + 6 in row 2. Here is the code. Given an array A with n elements and array B with m elements. With the following program, you can even print the sum of two numbers or three numbers up to N numbers. Given a circular array C of integers represented by A, find the maximum possible sum of a non-empty subarray of C. You need to find out the maximum sum such that no two chosen numbers are adjacent , vertically, diagonally (or) horizontally. The maximum product is formed by the (-10, -3) or (5, 6) pair. Calculates the weighted sum of two arrays. You can enter a value in the box labelled 'Person ID', which is the first number of a two-dimensional array, and then select one of two numbers in the Name/Profession box. =MAX(ROW(4:6)*SUM(ROW(1:3))) confirm with CTRL+SHIFT+ENTER. How to swap two numbers without using temporary variable? Write a program to print fibonacci series. In the given array, you need to find maximum sum of elements such that no two are adjacent (consecutive). To display sub array with maximum sum you should write code to hold the start and end value of the sub array with maximum sum. You can also return an array from a method. everything works except I need it to give me the max and min month (i. if 2,3,4, 5 is the given array, {4,5,2,3} is also a possible array like other two. Write a program to sort a	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
5 is the given array, {4,5,2,3} is also a possible array like other two. Write a program to sort a map by value. Though, the arrays whose size is a product of 2’s, 3’s, and 5’s (for example, 300 = 55322) are also processed quite efficiently. We can switch from one array to another array only at common elements. Function Description. Arr2[] = {1, 3, 3, 12, 2} then maximum result is obtained when we create following two subarrays − Subarr1[] = {2, 4, 3} and. You can refer to more than one array in a single SAS statement. Write a program to find the sum of the corresponding elements in 2 arrays. Maximum Sum of Two Non-Overlapping Subarrays. Inside SUM, the range resolves to an array of values. The restriction is that once you select a particular row,column value to add to your sum, no other values from that row or column may be used in calculating the sum. Please note that your values will almost certainly be different, depending both on the random number generator and the values of the array you created in Lab8. Finding the Average value of an Array. Mini-Max Sum Hackerrank. It could be the sum of total sum of the left child and max prefix sum of the right child. Find ways to calculate a target from elements of specified. Maximum Sub Array Practice: max_sub_array. Note that in the calculation of max4, we have passed a two dimensional array containing two rows of three elements as if it were a single dimensional array of six elements. 17171281366e-06 0. The first line of the input contains N,where N is the number of integers. Given two equally sized arrays (A, B) and N (size of both arrays). It should return a long integer that represents the maximum value of. We can switch from one array to another array only at common elements. The bottommost cell is A35. You can pass a two dimensional array to a method just as you pass a one dimensional array. Add solution to Pairs problem. Now let's think of another type of two dimensional array in which the shape of the array	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
problem. Now let's think of another type of two dimensional array in which the shape of the array is not square as shown below. More formally, if we write all (nC2) sub-arrays of array of size >=2 and find the sum of smallest and second smallest, then our answer will be maximum sum among them. Edit: given your comments if the initial array is fixed then you can use MMULT function like this. sum %2 [1] 1 1 [2] 2 0 [3] 3 1 [1,2] 3 1 [2,3] 5 1 [1,2,3] 6 0. Array Maximum Minimum value We can calculate maximum value among the elements of an array by using max function. ; The array formula lets the IF function test for multiple conditions in a single cell, and, when the data meets a condition, the array formula determines what data (event results) the MAX function will examine to find the best result. For example, if A is a matrix, then max(A,[],[1 2]) computes the maximum over all elements in A, since every element of a matrix is contained in the array slice defined by dimensions 1 and 2. Problem Description We have to write a program in C such that the program will allocate 2 one-dimensional arrays using malloc() call and then will do the addition and stores the result into 3rd array. We take a two dimensional array L of size count+1, sum/2+1. For example, given the array [-2,1,-3,4,-1,2,1,-5,4], the contiguous subarray [4,-1,2,1] has the largest sum = 6. min(int a, int b) and Math. Here we will be displaying the sub array. min () find the maximum and minimum value of the arguments, respectively. word 0:5 y:. The last 'n' integers correspond to the elements in the second array. pranathi chunduru. Hence there would be four different arrays in this case. For example, consider the array {-10, -3, 5, 6, -2}. Given an integer array of N elements, find the maximum sum contiguous subarray (containing at least one element). # sum of all elements in the array >np. Java represents a two-dimensional array as an array of arrays. For example, A = [−2, 1, −3, 4, −1, 2, 1, −5, 4] then	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
a two-dimensional array as an array of arrays. For example, A = [−2, 1, −3, 4, −1, 2, 1, −5, 4] then max sum=11 with the subarray [1, 4, 2, 4]. This example prints the maximum value in an array, and the subscript of that value:; Create a simple two-dimensional array: D = DIST (100); Print the maximum value in array D and its linear subscript: PRINT, 'Maximum value in array D is:', MAX (D, I) PRINT, 'The subscript of the maximum value is', I IDL Output Maximum value in array D is: 70. min, max, repmat, meshgrid, sum, cumsum, diff, prod, cumprod, filter 3. This program will help to understand the working of for loop, array, if statement and random numbers. Academia is the easiest way to share papers with millions of people across the world for free. I haven't gotten that far yet, I'm stuck just trying to print my two arrays, every time i try to print the first array it gives me the elements of the second array and it. Basic Operations ¶. com Great, I love this explanation. Reductions. The syntax of numpy. In this article we'll explore four plug and play functions that allow you to easily find certain values in an arrays of numbers. Improvement over Method-1 – O(n 2) Time. Output Format. I don't care if a document has many aliases matching my query. An index value of a Java two dimensional array starts at 0 and ends at n-1 where n is the size of a row or column. After partitioning, each subarray has their values changed to become the maximum value of that subarray. Two approaches: Simple approach is brute-force implementation but it will take O(n. Maximize array sum by concatenating corresponding elements of given two arrays Given two array A[] and B[] of the same length, the task is to find the maximum array sum that can be formed by joining the corresponding elements of the array in any order. Display the maximum K valid sum combinations from all the possible sum combinations. Timing Belt - $1,299. The code below will show how to display a maximum of 3 items from an	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
Timing Belt - $1,299. The code below will show how to display a maximum of 3 items from an integer array. Max sum in an array. Find the sum of the maximum sum path to reach from beginning of any array to end of any of the two arrays. As a precautionary health measure for our support specialists in light of COVID-19, we're operating with a limited team. Finding the maximum sum in two sorted arrays Given two sorted postive integer arrays A(n) and B(n) (let's say they are decreasingly sorted), we define a set S = {(a,b) \| a \in A and b \in B}. Mini-Max Sum - Problem from HackerRank and Solution Using python 2 Problem Statement: Given five positive integers, find the minimum and maximum values that can be calculated by summing exactly four of the five integers. The program allocates 2 one-dimentional arrays using malloc() call and then does the addition and stores the result into 3rd array. Method since it requires contiguous, it means that for each element, it has two situations that are in the subarray or not. Join 124,729,115 Academics and Researchers. You are given an array of integers with both positive and negative numbers. Subarr2[] = {3, 3, 12. Find the sum of numbers and represent it in array. The maximum special sum considering all non-empty subarrays of the array. In the Java programming language, a multidimensional array is an array whose components are themselves arrays. For example,$ \{35, 42, 5, 15, 27, 29\} $is a sorted array that has been circularly shifted$ k=2 $positions, while$ \{27, 29, 35, 42, 5, 15\} $has been shifted$ k=4 \$ positions. If only one array is supplied, SUMPRODUCT will simply sum the items in the array. C++ code to display contents of array. Find a Triplet having Maximum Product in an Array. Read 6 Integers from File, find sum, find average, and find Min/Max average. Write a program to find sum of each digit in the given number using recursion. Finding the Maximum value in an Array. Hi all, I'm looking for how to to the sum or union	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
recursion. Finding the Maximum value in an Array. Hi all, I'm looking for how to to the sum or union of two jagged array, I did it to simple array but do you have any idea how to do it for jagged array. For example, if an int. C Program to Find Maximum Element in Array - This program find maximum or largest element present in an array. We could also use other representations, such as an array containing two two-element arrays ([[76, 9], [4, 1]]) or an object with property names like "11" and "01", but the flat array is simple and makes the expressions that access the table pleasantly short. out [Optional] Alternate output array in which to place the. Sub Array with Maximum Sum – Kadane Algorithm is the best solution. Find the sum of the maximum sum path to reach from beginning of any array to end of any of the two arrays. In a two-dimensional Java array, we can use the code a[i] to refer to the ith row (which is a one-dimensional array). We can use an array as a deque with the following operations:. computes the sum of two matrices and then prints it. Search in Rotated Sorted Array II. As a precautionary health measure for our support specialists in light of COVID-19, we're operating with a limited team.	{ "domain": "lampedusasiamonoi.it", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.97594644290792, "lm_q1q2_score": 0.8963875270376713, "lm_q2_score": 0.9184802440252811, "openwebmath_perplexity": 681.2426012969111, "openwebmath_score": 0.3012574017047882, "tags": null, "url": "http://lampedusasiamonoi.it/yvgr/max-sum-of-2-arrays.html" }
# How to simplify or upperbound this summation? I am not a mathematician, so sorry for this trivial question. Is there a way to simplify or to upperbound the following summation: $$\sum_{i=1}^n{\exp{\left(-\frac{i^2}{\sigma^2}\right)}}.$$ Can I use geometric series? EDIT: I have difficulty because of the power $2$, i.e if the summation would be $\sum\limits_{i=1}^n{\exp{\left(-\frac{i}{\sigma^2}\right)}}$ then it would be easy to apply geometric series! • Don't be sorry for the question! The only way to get better as mathematician is to ask questions. (See also the first annotation to this post) – Jacob Manaker Sep 18 at 16:18 • Seems that you could try the integral $\int_0^\infty \exp(-x^2)\,\mathrm dx$ to bound that. – xbh Sep 18 at 16:26 • tinyurl.com/y76n9loh (wolframalpha) gives a closed form for the infinite sum involving the elliptic theta function. – barrycarter Sep 19 at 23:46 Alternative: Since $f(x) = \exp(-x^2/\sigma^2) \searrow 0$, we can write $\DeclareMathOperator{\diff}{\,d\!}$ \begin{align} &\sum_1^n \exp\left(-\frac {j^2}{\sigma^2}\right) \\ &= \sum_1^n \int_{j-1}^j \exp\left(-\frac {j^2}{\sigma^2}\right)\diff x \\ &\leqslant \sum_1^n \int_{j-1}^j \exp\left(-\frac {x^2}{\sigma^2}\right) \diff x \\ &=\sigma \int_0^n \exp\left(-\frac {x^2}{\sigma^2}\right) \diff \left(\frac x \sigma \right)\\ &= \sigma \int_0^{n/\sigma} \exp(-x^2)\diff x\\ &\leqslant \sigma \int_0^{+\infty}\exp(-x^2)\diff x\\ &= \frac \sigma 2 \sqrt \pi \end{align} • Nicely done! Sometimes simplest is best. – Jacob Manaker Sep 18 at 17:54 • brilliant answer, thanks – user8003788 Sep 19 at 7:39 • @JacobManaker Thanks for compliment! – xbh Sep 19 at 7:43 TL;DR: three relatively easy bounds are the numbered equations below.	{ "domain": "stackexchange.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.982287698185481, "lm_q1q2_score": 0.8962748204939407, "lm_q2_score": 0.9124361652391386, "openwebmath_perplexity": 522.4824194582968, "openwebmath_score": 0.9999333620071411, "tags": null, "url": "https://math.stackexchange.com/questions/2921700/how-to-simplify-or-upperbound-this-summation" }
TL;DR: three relatively easy bounds are the numbered equations below. You cannot directly apply the formula for the geometric series for the reason mentioned in your edit. But note that $i\geq1$, so we have $$\sum_{i=1}^n{\exp{\left(-\frac{i^2}{\sigma^2}\right)}}\leq\sum_{i=1}^n{\exp{\left(-\frac{i\cdot1}{\sigma^2}\right)}}$$ The latter, of course, is a geometric sum. Taking the sum over all $i$ (including $i=0$), we get $$(1-e^{-\sigma^{-2}})^{-1} \tag{1} \label{eqn:first}$$ The calculation for finitely many terms isn't much harder, and only differs by an exponentially decreasing factor. If this isn't a strong enough bound, there are other techniques. If $n<\sigma$, then we can get very far elementarily. Note that $e^x\geq x+1$; dividing each side, we get $$e^{-x}\leq(1+x)^{-1}=\sum_{k=0}^{\infty}{(-x)^k}$$ if $\|x\|<1$. Taking $x=\left(\frac{i}{\sigma}\right)^2$, we thus obtain \begin{align} \sum_{i=1}^n{e^{-\frac{i^2}{\sigma^2}}}&\leq\sum_{i=1}^n{\sum_{k=0}^{\infty}{\left(-\left(\frac{i}{\sigma}\right)^2\right)^k}} \\ &=\sum_{k=0}^{\infty}{(-1)^k\sum_{i=1}^n{\left(\frac{i}{\sigma}\right)^{2k}}} \tag{} \label{eqn:star} \end{align*} (We can interchange sums because one is finite.) Now, for all $k$, the function $\left(\frac{\cdot}{\sigma}\right)^{2k}$ is increasing on $[0,\infty)$; we thus have $$\int_0^n{\left(\frac{i}{\sigma}\right)^{2k}\,di}\leq\sum_{i=1}^n{\left(\frac{i}{\sigma}\right)^{2k}}\leq\left(\frac{n}{\sigma}\right)^{2k}+\int_1^n{\left(\frac{i}{\sigma}\right)^{2k}\,di}$$ Evaluating the integrals and simplifying, we have $$0\leq\sum_{i=1}^n{\left(\frac{i}{\sigma}\right)^{2k}}-\frac{n}{2k+1}\left(\frac{n}{\sigma}\right)^{2k}\leq\left(\frac{n}{\sigma}\right)^{2k}\left(1-\frac{1}{(2k+1)n^{2k}}\right)$$	{ "domain": "stackexchange.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.982287698185481, "lm_q1q2_score": 0.8962748204939407, "lm_q2_score": 0.9124361652391386, "openwebmath_perplexity": 522.4824194582968, "openwebmath_score": 0.9999333620071411, "tags": null, "url": "https://math.stackexchange.com/questions/2921700/how-to-simplify-or-upperbound-this-summation" }
Substituting into $\eqref{eqn:star}$, we get \begin{align} \sum_{i=1}^n{e^{-\frac{i^2}{\sigma^2}}}&\leq\sum_{k=0}^{\infty}{\frac{(-1)^kn}{2k+1}\left(\frac{n}{\sigma}\right)^{2k}}-\sum_{j=0}^{\infty}{\left(\frac{n}{\sigma}\right)^{4j+2}\left(1-\frac{1}{(4j+3)n^{4j+2}}\right)} \\ &\leq\sum_{k=0}^{\infty}{\frac{(-1)^kn}{2k+1}\left(\frac{n}{\sigma}\right)^{2k}}-\sum_{j=0}^{\infty}{\left(\frac{n}{\sigma}\right)^{4j+2}} \\ &=\sigma\tan^{-1}{\left(\frac{n}{\sigma}\right)}-\frac{\left(\frac{n}{\sigma}\right)^2}{1-\left(\frac{n}{\sigma}\right)^4}\hspace{4em}(n<\sigma) \tag{2} \end{align} Finally, for the general case we can achieve a slight improvement on $\eqref{eqn:first}$ via the theory of majorization. $\{x_i\}_{i=1}^n\mapsto\sum_{i=1}^n{\exp{\left(-\frac{x_i}{\sigma^2}\right)}}$ is convex and symmetric in its arguments, hence Schur-convex. Let $b_i=i^2$ and $a_i=\left(\frac{2n-1}{3}\right)i$. Clearly, for all $m\leq n$, we have $$\sum_{i=1}^m{a_i}=\frac{m(m-1)}{2}\cdot\frac{2n-1}{3}\geq\frac{m(m-1)(2m-1)}{6}=\sum_{i=1}^m{b_i}$$ with equality if $m=n$. Thus $\vec{a}$ majorizes $\vec{b}$, so \begin{align} \sum_{i=1}^n{\exp{\left(-\frac{i^2}{\sigma^2}\right)}}&=\sum_{i=1}^n{\exp{\left(-\frac{b_i}{\sigma^2}\right)}} \\ &\leq\sum_{i=1}^n{\exp{\left(-\frac{a_i}{\sigma^2}\right)}} \\ &=\sum_{i=1}^n{\exp{\left(-\frac{(2n-1)i}{3\sigma^2}\right)}} \\ &\leq\sum_{i=0}^{\infty}{\exp{\left(-\frac{(2n-1)i}{3\sigma^2}\right)}} \\ &\leq\left(1-\exp{\left(\frac{2n-1}{3\sigma^2}\right)}\right)^{-1} \tag{3} \end{align} • Great detailed work..I considered previously the first answer but I thought it would be better if I can get a stronger bound.Thanks a lot – user8003788 Sep 19 at 7:42 There's a rather trivial upper bound that $\frac{-i^2}{\sigma^2}$ is negative, so exponentiating it results in a number less than 1, so the sum is at most $n$. If you want a constant upper bound, you can upper bound it with the geometric series.	{ "domain": "stackexchange.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.982287698185481, "lm_q1q2_score": 0.8962748204939407, "lm_q2_score": 0.9124361652391386, "openwebmath_perplexity": 522.4824194582968, "openwebmath_score": 0.9999333620071411, "tags": null, "url": "https://math.stackexchange.com/questions/2921700/how-to-simplify-or-upperbound-this-summation" }
The matter is that $e^{-(x/ \sigma)^2}$, in the range $0 \le x < \approx \sigma$ is very steep. So, unless $\sigma$ is quite high, you cannot get a good approximation by the integral. But of course everything depends on the parameters into play and on the accuracy required. Hint : For general values of $n$ and $\sigma$ it might be interesting to take advantage of the fact that the Fourier Transform of a Gaussian is a Gaussian itself. Then you are taking the signal $e^{-\, (t/ \sigma)^2}$, windowing it between $0 \le t \le n/ \sigma$, taking $n$ samples of it, and after that you are taking $n$ times the average. All these operations have a simple translation into the frequency domain. However I do not go further not knowing whether you are acknowledged in this field, and keep this as a hint. Also, might be interesting this identity $$\sum\limits_{k \in \mathbb Z} {\exp \left( { - \pi \left( {k/c} \right)^2 } \right)} = c\sum\limits_{k \in \mathbb Z} {\exp \left( { - \pi \left( {k\,c} \right)^2 } \right)}$$ reported at the end of the Properties paragraph in this wikipedia article.	{ "domain": "stackexchange.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.982287698185481, "lm_q1q2_score": 0.8962748204939407, "lm_q2_score": 0.9124361652391386, "openwebmath_perplexity": 522.4824194582968, "openwebmath_score": 0.9999333620071411, "tags": null, "url": "https://math.stackexchange.com/questions/2921700/how-to-simplify-or-upperbound-this-summation" }
# Tag Archives: polynomial ## Infinite Ways to an Infinite Geometric Sum One of my students, K, and I were reviewing Taylor Series last Friday when she asked for a reminder why an infinite geometric series summed to $\displaystyle \frac{g}{1-r}$ for first term g and common ratio r when $\left\| r \right\| < 1$. I was glad she was dissatisfied with blind use of a formula and dove into a familiar (to me) derivation. In the end, she shook me free from my routine just as she made sure she didn’t fall into her own. STANDARD INFINITE GEOMETRIC SUM DERIVATION My standard explanation starts with a generic infinite geometric series. $S = g+g\cdot r+g\cdot r^2+g\cdot r^3+...$ (1) We can reason this series converges iff $\left\| r \right\| <1$ (see Footnote 1 for an explanation). Assume this is true for (1). Notice the terms on the right keep multiplying by r. The annoying part of summing any infinite series is the ellipsis (…). Any finite number of terms always has a finite sum, but that simply written, but vague ellipsis is logically difficult. In the geometric series case, we might be able to handle the ellipsis by aligning terms in a similar series. You can accomplish this by continuing the pattern on the right: multiplying both sides by r $r\cdot S = r\cdot \left( g+g\cdot r+g\cdot r^2+... \right)$ $r\cdot S = g\cdot r+g\cdot r^2+g\cdot r^3+...$ (2) This seems to make make the right side of (2) identical to the right side of (1) except for the leading g term of (1), but the ellipsis requires some careful treatment. Footnote 2 explains how the ellipses of (1) and (2) are identical. After that is established, subtracting (2) from (1), factoring, and rearranging some terms leads to the infinite geometric sum formula. $(1)-(2) = S-S\cdot r = S\cdot (1-r)=g$ $\displaystyle S=\frac{g}{1-r}$ STUDENT PREFERENCES	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
$(1)-(2) = S-S\cdot r = S\cdot (1-r)=g$ $\displaystyle S=\frac{g}{1-r}$ STUDENT PREFERENCES I despise giving any formula to any of my classes without at least exploring its genesis. I also allow my students to use any legitimate mathematics to solve problems so long as reasoning is justified. In my experiences, about half of my students opt for a formulaic approach to infinite geometric sums while an equal number prefer the quick “multiply-by-r-and-subtract” approach used to derive the summation formula. For many, apparently, the dynamic manipulation is more meaningful than a static rule. It’s very cool to watch student preferences at play. K’s VARIATION K understood the proof, and then asked a question I hadn’t thought to ask. Why did we have to multiply by r? Could multiplication by $r^2$ also determine the summation formula? I had three nearly simultaneous thoughts followed quickly by a fourth. First, why hadn’t I ever thought to ask that? Second, geometric series for $\left\| r \right\|<1$ are absolutely convergent, so K’s suggestion should work. Third, while the formula would initially look different, absolute convergence guaranteed that whatever the “$r^2$ formula” looked like, it had to be algebraically equivalent to the standard form. While I considered those conscious questions, my math subconscious quickly saw the easy resolution to K’s question and the equivalence from Thought #3. Multiplying (1) by $r^2$ gives $r^2 \cdot S = g\cdot r^2 + g\cdot r^3 + ...$ (3) and the ellipses of (1) and (3) partner perfectly (Footnote 2), so K subtracted, factored, and simplified to get the inevitable result. $(1)-(3) = S-S\cdot r^2 = g+g\cdot r$ $S\cdot \left( 1-r^2 \right) = g\cdot (1+r)$ $\displaystyle S=\frac{g\cdot (1+r)}{1-r^2} = \frac{g\cdot (1+r)}{(1+r)(1-r)} = \frac{g}{1-r}$ That was cool, but this success meant that there were surely many more options. EXTENDING	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
That was cool, but this success meant that there were surely many more options. EXTENDING Why stop at multiplying by r or $r^2$? Why not multiply both sides of (1) by a generic $r^N$ for any natural number N? That would give $r^N \cdot S = g\cdot r^N + g\cdot r^{N+1} + ...$ (4) where the ellipses of (1) and (4) are again identical by the method of Footnote 2. Subtracting (4) from (1) gives $(1)-(4) = S-S\cdot r^N = g+g\cdot r + g\cdot r^2+...+ g\cdot r^{N-1}$ $S\cdot \left( 1-r^N \right) = g\cdot \left( 1+r+r^2+...+r^{N-1} \right)$ (5) There are two ways to proceed from (5). You could recognize the right side as a finite geometric sum with first term 1 and ratio r. Substituting that formula and dividing by $\left( 1-r^N \right)$ would give the general result. Alternatively, I could see students exploring $\left( 1-r^N \right)$, and discovering by hand or by CAS that $(1-r)$ is always a factor. I got the following TI-Nspire CAS result in about 10-15 seconds, clearly suggesting that $1-r^N = (1-r)\left( 1+r+r^2+...+r^{N-1} \right)$. (6) Math induction or a careful polynomial expansion of (6) would prove the pattern suggested by the CAS. From there, dividing both sides of (5) by $\left( 1-r^N \right)$ gives the generic result. $\displaystyle S = \frac{g\cdot \left( 1+r+r^2+...+r^{N-1} \right)}{\left( 1-r^N \right)}$ $\displaystyle S = \frac{g\cdot \left( 1+r+r^2+...+r^{N-1} \right) }{(1-r) \cdot \left( 1+r+r^2+...+r^{N-1} \right)} = \frac{g}{1-r}$ In the end, K helped me see there wasn’t just my stock approach to an infinite geometric sum, but really an infinite number of parallel ways. Nice. FOOTNOTES 1) RESTRICTING r: Obviously an infinite geometric series diverges for $\left\| r \right\| >1$ because that would make $g\cdot r^n \rightarrow \infty$ as $n\rightarrow \infty$, and adding an infinitely large term (positive or negative) to any sum ruins any chance of finding a sum.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
For $r=1$, the sum converges iff $g=0$ (a rather boring series). If $g \ne 0$ , you get a sum of an infinite number of some nonzero quantity, and that is always infinite, no matter how small or large the nonzero quantity. The last case, $r=-1$, is more subtle. For $g \ne 0$, this terms of this series alternate between positive and negative g, making the partial sums of the series add to either g or 0, depending on whether you have summed an even or an odd number of terms. Since the partial sums alternate, the overall sum is divergent. Remember that series sums and limits are functions; without a single numeric output at a particular point, the function value at that point is considered to be non-existent. 2) NOT ALL INFINITIES ARE THE SAME: There are two ways to show two groups are the same size. The obvious way is to count the elements in each group and find out there is the same number of elements in each, but this works only if you have a finite group size. Alternatively, you could a) match every element in group 1 with a unique element from group 2, and b) match every element in group 2 with a unique element from group 1. It is important to do both steps here to show that there are no left-over, unpaired elements in either group. So do the ellipses in (1) and (2) represent the same sets? As the ellipses represent sets with an infinite number of elements, the first comparison technique is irrelevant. For the second approach using pairing, we need to compare individual elements. For every element in the ellipsis of (1), obviously there is an “partner” in (2) as the multiplication of (1) by r visually shifts all of the terms of the series right one position, creating the necessary matches.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
Students often are troubled by the second matching as it appears the ellipsis in (2) contains an “extra term” from the right shift. But, for every specific term you identify in (2), its identical twin exists in (1). In the weirdness of infinity, that “extra term” appears to have been absorbed without changing the “size” of the infinity. Since there is a 1:1 mapping of all elements in the ellipses of (1) and (2), you can conclude they are identical, and their difference is zero. ## Probability, Polynomials, and Sicherman Dice Three years ago, I encountered a question on the TI-Nspire Google group asking if there was a way to use CAS to solve probability problems. The ideas I pitched in my initial response and follow-up a year later (after first using it with students in a statistics class) have been thoroughly re-confirmed in my first year teaching AP Statistics. I’ll quickly re-share them below before extending the concept with ideas I picked up a couple weeks ago from Steve Phelps’ session on Probability, Polynomials, and CAS at the 64th annual OCTM conference earlier this month in Cleveland, OH. BINOMIALS: FROM POLYNOMIALS TO SAMPLE SPACES Once you understand them, binomial probability distributions aren’t that difficult, but the initial conjoining of combinatorics and probability makes this a perennially difficult topic for many students. The standard formula for the probability of determining the chances of K successes in N attempts of a binomial situation where p is the probability of a single success in a single attempt is no less daunting: $\displaystyle \left( \begin{matrix} N \\ K \end{matrix} \right) p^K (1-p)^{N-K} = \frac{N!}{K! (N-K)!} p^K (1-p)^{N-K}$	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
But that is almost exactly the same result one gets by raising binomials to whole number powers, so why not use a CAS to expand a polynomial and at least compute the $\displaystyle \left( \begin{matrix} N \\ K \end{matrix} \right)$ portion of the probability? One added advantage of using a CAS is that you could use full event names instead of abbreviations, making it even easier to identify the meaning of each event. The TI-Nspire output above shows the entire sample space resulting from flipping a coin 6 times. Each term is an event. Within each term, the exponent of each variable notes the number of times that variable occurs and the coefficient is the number of times that combination occurs. The overall exponent in the expand command is the number of trials. For example, the middle term– $20\cdot heads^3 \cdot tails^3$ –says that there are 20 ways you could get 3 heads and 3 tails when tossing a coin 6 times. The last term is just $tails^6$, and its implied coefficient is 1, meaning there is just one way to flip 6 tails in 6 tosses. The expand command makes more sense than memorized algorithms and provides context to students until they gain a deeper understanding of what’s actually going on. FROM POLYNOMIALS TO PROBABILITY Still using the expand command, if each variable is preceded by its probability, the CAS result combines the entire sample space AND the corresponding probability distribution function. For example, when rolling a fair die four times, the distribution for 1s vs. not 1s (2, 3, 4, 5, or 6) is given by	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
The highlighted term says there is a 38.58% chance that there will be exactly one 1 and any three other numbers (2, 3, 4, 5, or 6) in four rolls of a fair 6-sided die. The probabilities of the other four events in the sample space are also shown. Within the TI-Nspire (CAS or non-CAS), one could use a command to give all of these probabilities simultaneously (below), but then one has to remember whether the non-contextualized probabilities are for increasing or decreasing values of which binomial outcome. Particularly early on in their explorations of binomial probabilities, students I’ve taught have shown a very clear preference for the polynomial approach, even when allowed to choose any approach that makes sense to them. TAKING POLYNOMIALS FROM ONE DIE TO MANY Given these earlier thoughts, I was naturally drawn to Steve Phelps “Probability, Polynomials, and CAS” session at the November 2014 OCTM annual meeting in Cleveland, OH. Among the ideas he shared was using polynomials to create the distribution function for the sum of two fair 6-sided dice. My immediate thought was to apply my earlier ideas. As noted in my initial post, the expansion approach above is not limited to binomial situations. My first reflexive CAS command in Steve’s session before he share anything was this. By writing the outcomes in words, the CAS interprets them as variables. I got the entire sample space, but didn’t learn gain anything beyond a long polynomial. The first output– $five^2$ –with its implied coefficient says there is 1 way to get 2 fives. The second term– $2\cdot five \cdot four$ –says there are 2 ways to get 1 five and 1 four. Nice that the technology gives me all the terms so quickly, but it doesn’t help me get a distribution function of the sum. I got the distributions of the specific outcomes, but the way I defined the variables didn’t permit sum of their actual numerical values. Time to listen to the speaker.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
He suggested using a common variable, X, for all faces with the value of each face expressed as an exponent. That is, a standard 6-sided die would be represented by $X^1+X^2+ X^3+X^4+X^5+X^6$ where the six different exponents represent the numbers on the six faces of a typical 6-sided die. Rolling two such dice simultaneously is handled as I did earlier with the binomial cases. NOTE: Exponents are handled in TWO different ways here. 1) Within a single polynomial, an exponent is an event value, and 2) Outside a polynomial, an exponent indicates the number of times that polynomial is applied within the specific event. Coefficients have the same meaning as before. Because the variables are now the same, when specific terms are multiplied, their exponents (face values) will be added–exactly what I wanted to happen. That means the sum of the faces when you roll two dice is determined by the following. Notice that the output is a single polynomial. Therefore, the exponents are the values of individual cases. For a couple examples, there are 3 ways to get a sum of 10 $\left( 3 \cdot x^{10} \right)$, 2 ways to get a sum of 3 $\left( 2 \cdot x^3 \right)$, etc. The most commonly occurring outcome is the term with the largest coefficient. For rolling two standard fair 6-sided dice, a sum of 7 is the most common outcome, occurring 6 times $\left( 6 \cdot x^7 \right)$. That certainly simplifies the typical 6×6 tables used to compute the sums and probabilities resulting from rolling two dice. While not the point of Steve’s talk, I immediately saw that technology had just opened the door to problems that had been computationally inaccessible in the past. For example, what is the most common sum when rolling 5 dice and what is the probability of that sum? On my CAS, I entered this.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
In the middle of the expanded polynomial are two terms with the largest coefficients, $780 \cdot x^{18}$ and $780 \cdot x^{19}$, meaning a sums of 17 and 18 are the most common, equally likely outcomes when rolling 5 dice. As there are $6^5=7776$ possible outcomes when rolling a die 5 times, the probability of each of these is $\frac{780}{7776} \approx 0.1003$, or about 10.03% chance each for a sum of 17 or 18. This can be verified by inserting the probabilities as coefficients before each term before CAS expanding. With thought, this shouldn’t be surprising as the expected mean value of rolling a 6-sided die many times is 3.5, and $5 \cdot 3.5 = 17.5$, so the integers on either side of 17.5 (17 & 18) should be the most common. Technology confirms intuition. ROLLING DIFFERENT DICE SIMULTANEOUSLY What is the distribution of sums when rolling a 4-sided and a 6-sided die together? No problem. Just multiply two different polynomials, one representative of each die. The output shows that sums of 5, 6, and 7 would be the most common, each occurring four times with probability $\frac{1}{6}$ and together accounting for half of all outcomes of rolling these two dice together. A BEAUTIFUL EXTENSION–SICHERMAN DICE My most unexpected gain from Steve’s talk happened when he asked if we could get the same distribution of sums as “normal” 6-sided dice, but from two different 6-sided dice. The only restriction he gave was that all of the faces of the new dice had to have positive values. This can be approached by realizing that the distribution of sums of the two normal dice can be found by multiplying two representative polynomials to get $x^{12}+2x^{11}+3x^{10}+4x^9+5x^8+6x^7+5x^6+4x^5+3x^4+2x^3+x^2$. Restating the question in the terms of this post, are there two other polynomials that could be multiplied to give the same product? That is, does this polynomial factor into other polynomials that could multiply to the same product? A CAS factor command gives	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
Any rearrangement of these eight (four distinct) sub-polynomials would create the same distribution as the sum of two dice, but what would the the separate sub-products mean in terms of the dice? As a first example, what if the first two expressions were used for one die (line 1 below) and the two squared trinomials comprised a second die (line 2)? Line 1 actually describes a 4-sided die with one face of 4, two faces with 3s, and one face of 2. Line 2 describes a 9-sided die (whatever that is) with one face of 8, two faces of 6, three faces of 4, two faces of 2, and one face with a 0 ( $1=1 \cdot x^0$). This means rolling a 4-sided and a 9-sided die as described would give exactly the same sum distribution. Cool, but not what I wanted. Now what? Factorization gave four distinct sub-polynomials, each with multitude 2. One die could contain 0, 1, or 2 of each of these with the remaining factors on the other die. That means there are $3^4=81$ different possible dice combinations. I could continue with a trail-and-error approach, but I wanted to be more efficient and elegant. What follows is the result of thinking about the problem for a while. Like most math solutions to interesting problems, ultimate solutions are typically much cleaner and more elegant than the thoughts that went into them. Problem solving is a messy–but very rewarding–business. SOLUTION Here are my insights over time: 1) I realized that the $x^2$ term would raise the power (face values) of the desired dice, but would not change the coefficients (number of faces). Because Steve asked for dice with all positive face values. That meant each desired die had to have at least one x to prevent non-positive face values.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
2) My first attempt didn’t create 6-sided dice. The sums of the coefficients of the sub-polynomials determined the number of sides. That sum could also be found by substituting $x=1$ into the sub-polynomial. I want 6-sided dice, so the final coefficients must add to 6. The coefficients of the factored polynomials of any die individually must add to 2, 3, or 6 and have a product of 6. The coefficients of $(x+1)$ add to 2, $\left( x^2+x+1 \right)$ add to 3, and $\left( x^2-x+1 \right)$ add to 1. The only way to get a polynomial coefficient sum of 6 (and thereby create 6-sided dice) is for each die to have one $(x+1)$ factor and one $\left( x^2+x+1 \right)$ factor. 3) That leaves the two $\left( x^2-x+1 \right)$ factors. They could split between the two dice or both could be on one die, leaving none on the other. We’ve already determined that each die already had to have one each of the x, $(x+1)$, and $\left( x^2+x+1 \right)$ factors. To also split the $\left( x^2-x+1 \right)$ factors would result in the original dice: Two normal 6-sided dice. If I want different dice, I have to load both of these factors on one die. That means there is ONLY ONE POSSIBLE alternative for two 6-sided dice that have the same sum distribution as two normal 6-sided dice. One die would have single faces of 8, 6, 5, 4, 3, and 1. The other die would have one 4, two 3s, two 2s, and one 1. And this is exactly the result of the famous(?) Sicherman Dice. If a 0 face value was allowed, shift one factor of x from one polynomial to the other. This can be done two ways. The first possibility has dice with faces {9, 7, 6, 5, 4, 2} and {3, 2, 2, 1, 1, 0}, and the second has faces {7, 5, 4, 3, 2, 0} and {5, 4, 4, 3, 3, 2}, giving the only other two non-negative solutions to the Sicherman Dice.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
Both of these are nothing more than adding one to all faces of one die and subtracting one from from all faces of the other. While not necessary to use polynomials to compute these, they are equivalent to multiplying the polynomial of one die by x and the other by $\frac{1}{x}$ as many times as desired. That means there are an infinite number of 6-sided dice with the same sum distribution as normal 6-sided dice if you allow the sides to have negative faces. One of these is corresponding to a pair of Sicherman Dice with faces {6, 4, 3, 2, 1, -1} and {1,5,5,4,4,3}. CONCLUSION: There are other very interesting properties of Sicherman Dice, but this is already a very long post. In the end, there are tremendous connections between probability and polynomials that are accessible to students at the secondary level and beyond. And CAS keeps the focus on student learning and away from the manipulations that aren’t even the point in these explorations. Enjoy. ## Number Bases and Polynomials About a month ago, I was working with our 5th grade math teacher to develop some extension activities for some students in an unleveled class. The class was exploring place value, and I suggested that some might be ready to explore what happens when you allow the number base to be something other than 10. A few students had some fun learning to use their basic four algorithms in other number bases, but I made an even deeper connection. When writing something like 512 in expanded form ($5\cdot 10^2+1\cdot 10^1+2\cdot 10^0$), I realized that if the 10 was an x, I’d have a polynomial. I’d recognized this before, but this time I wondered what would happen if I applied basic math algorithms to polynomials if I wrote them in a condensed numerical form, not their standard expanded form. That is, could I do basic algebra on $5x^2+x+2$ if I thought of it as $512_x$–a base-x “number”? (To avoid other confusion later, I read this as “five one two base-x“.)	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
Following are some examples I played with to convince myself how my new notation would work. I’m not convinced that this will ever lead to anything, but following my “what ifs” all the way to infinite series was a blast. Read on! If I wanted to add $(3x+5)$$(2x^2+4x+1)$, I could think of it as $35_x+241_x$ and add the numbers “normally” to get $276_x$ or $2x^2+7x+6$. Notice that each power of x identifies a “place value” for its characteristic coefficient. If I wanted to add $3x-7$ to itself, I had to adapt my notation a touch. The “units digit” is a negative number, but since the number base, x, is unknown (or variable), I ended up saying $3x-7=3(-7)_x$. The parentheses are used to contain multiple characters into a single place value. Then, $(3x-7)+(3x-7)$ becomes $3(-7)_x+3(-7)_x=6(-14)_x$ or $6x-14$. Notice the expanding parentheses containing the base-x units digit. The last example also showed me that simple multiplication would work. Adding $3x-7$ to itself is equivalent to multiplying $2\cdot (3x-7)$. In base-x, that is $2\cdot 3(-7)_x$. That’s easy! Arguably, this might be even easier that doubling a number when the number base is known. Without interactions between the coefficients of different place values, just double each digit to get $6(-14)_x=6x-14$, as before. What about $(x^2+7)+(8x-9)$? That’s equivalent to $107_x+8(-9)_x$. While simple, I’ll solve this one by stacking. and this is $x^2+8x-2$. As with base-10 numbers, the use of 0 is needed to hold place values exactly as I needed a 0 to hold the $x^1$ place for $x^2+7$. Again, this could easily be accomplished without the number base conversion, but how much more can we push these boundaries? Level 3–Multiplication & Powers: Compute $(8x-3)^2$. Stacking again and using a modification of the multiply-and-carry algorithm I learned in grade school, I got and this is equivalent to $64x^2-48x+9$. All other forms of polynomial multiplication work just fine, too.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
All other forms of polynomial multiplication work just fine, too. From one perspective, all of this shifting to a variable number base could be seen as completely unnecessary. We already have acceptably working algorithms for addition, subtraction, and multiplication. But then, I really like how this approach completes the connection between numerical and polynomial arithmetic. The rules of math don’t change just because you introduce variables. For some, I’m convinced this might make a big difference in understanding. I also like how easily this extends polynomial by polynomial multiplication far beyond the bland monomial and binomial products that proliferate in virtually all modern textbooks. Also banished here is any need at all for banal FOIL techniques. Level 4–Division: What about $x^2+x-6$ divided by $x+3$? In base-x, that’s $11(-6)_x \div 13_x$. Remembering that there is no place value carrying possible, I had to be a little careful when setting up my computation. Focusing only on the lead digits, 1 “goes into” 1 one time. Multiplying the partial quotient by the divisor, writing the result below and subtracting gives Then, 1 “goes into” -2 negative two times. Multiplying and subtracting gives a remainder of 0. thereby confirming that $x+3$ is a factor of $x^2+x-6$, and the other factor is the quotient, $x-2$. Perhaps this could be used as an alternative to other polynomial division algorithms. It is somewhat similar to the synthetic division technique, without its significant limitations: It is not limited to linear divisors with lead coefficients of one. For $(4x^3-5x^2+7) \div (2x^2-1)$, think $4(-5)07_x \div 20(-1)_x$. Stacking and dividing gives So $\displaystyle \frac{4x^3-5x^2+7}{2x^2-1}=2x-2.5+\frac{2x+4.5}{2x^2-1}$. CONCLUSION	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
So $\displaystyle \frac{4x^3-5x^2+7}{2x^2-1}=2x-2.5+\frac{2x+4.5}{2x^2-1}$. CONCLUSION From all I’ve been able to tell, converting polynomials to their base-x number equivalents enables you to perform all of the same arithmetic computations. For division in particular, it seems this method might even be a bit easier. In my next post, I push the exploration of these base-x numbers into infinite series. ## Extending graph control This article takes my idea from yesterday’s post about using $g(x)=\sqrt \frac{\left \| x \right \|}{x}$ to control the appearance of a graph and extends it in two ways. • Part I below uses Desmos to graph $y=(x+2)^3x^2(x-1)$ from the left and right simultaneously • Part II was inspired by my Twitter colleague John Burk who asked if this control could be extended in a different direction. Part I: Simultaneous Control When graphing polynomials like $y=(x+2)^3x^2(x-1)$, I encourage my students to use both its local behavior (cubic root at $x=-2$, quadratic root at $x=0$, and linear root at $x=1$) and its end behavior (6th degree polynomial with a positive lead coefficient means $y\rightarrow +\infty$ as $x\rightarrow\pm\infty$). To start graphing, I suggest students plot points on the x-intercepts and then sketch arrows to indicate the end behavior. In the past, this was something we did on paper, but couldn’t get technology to replicate it live–until this idea. In class last week, I used a minor extension of yesterday’s idea to control a graph’s appearance from the left and right simultaneously. Yesterday’s post suggested multiplying by $\sqrt \frac{\left \| a-x \right \|}{a-x}$ to show the graph of a function from the left for $x. Creating a second graph multiplied by $\sqrt \frac{\left \| x-b \right \|}{x-b}$ gives a graph of your function from the right for $b. The following images show the polynomial’s graph developing in a few stages. You can access the Desmos file here.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
First graph the end behavior (pull the a and b sliders in a bit to see just the ends of the graph) and plot points at the x-intercepts. From here, you could graph left-to-right or right-to-left. I’ll come in from the right to show the new right side controller. The root at $x=1$ is linear, so decreasing the b slider to just below 1 shows this. Continuing from the right, the next root is a bounce at $x=0$, as shown by decreasing the b slider below 0. Notice that this forces a relative minimum for some $0. Just because it’s possible, I’ll now show the cubic intercept at $x=2$ by increasing the a slider above 2. All that remains is to connect the two sides of the graph, creating one more relative minimum in $-2. The same level of presentation control can be had for any function’s graph. Part II: Vertical Control I hadn’t thought to extend this any further until my colleague asked if a graph could be controlled up and down instead of left and right. My guess is that the idea hadn’t occurred to me because I typically think about controlling a function through its domain. Even so, a couple minor adjustments accomplished it. Click here to see a vertical control of the graph of $y=x^3$ from above and below. Enjoy. ## Quadratics, Statistics, Symmetry, and Tranformations A problem I assigned my precalculus class this past Thursday ended up with multiple solutions by the time we finished. Huzzah for student creativity! The question: Find equations for all polynomial functions, $y=f(x)$, of degree $\le 2$ for which $f(0)=f(1)=2$ and $f(3)=0$. After they had worked on this (along with several variations on the theme), four very different ways of thinking about this problem emerged. All were valid and even led to a lesson I hadn’t planned–proving that, even though they looked different algebraically, all were equivalent. I present their approaches (and a few extras) in the order they were offered in our post-solving debriefing.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
The commonality among the approaches was their recognition that 3 non-collinear points uniquely define a vertical parabola, so they didn’t need to worry about polynomials of degree 0 or 1. (They haven’t yet heard about rotated curves that led to my earlier post on rotated quadratics.) Solution 1–Regression: Because only 3 points were given, a quadratic regression would derive a perfectly fitting quadratic equation. Using their TI-Nspire CASs, they started by entering the 3 ordered pairs in a Lists&Spreadsheets window. Most then went to a Calculator window to compute a quadratic regression. Below, I show the same approach using a Data&Statistics window instead so I could see simultaneously the curve fit and the given points. The decimals were easy enough to interpret, so even though they were presented in decimal form, these students reported $y=-\frac{1}{3}x^2+\frac{1}{3}x+2$. For a couple seconds after this was presented, I honestly felt a little cheated. I was hoping they would tap the geometric or algebraic properties of quadratics to get their equations. But I then I remembered that I clearly hadn’t make that part of my instructions. After my initial knee-jerk reaction, I realized this group of students had actually done exactly what I explicitly have been encouraging them to do: think freely and take advantage of every tool they have to find solutions. Nothing in the problem statement suggested technology or regressions, so while I had intended a more geometric approach, I realized I actually owed these students some kudos for a very creative, insightful, and technology-based solution. This and Solution 2 were the most frequently chosen approaches.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
Solution 2–Systems: Equations of quadratic functions are typically presented in standard, factored, or vertex form. Since neither two zeros nor the vertex were explicitly given, the largest portion of the students used the standard form, $y=a\cdot x^2+b\cdot x+c$ to create a 3×3 system of equations. Some solved this by hand, but most invoked a CAS solution. Notice the elegance of the solve command they used, working from the generic polynomial equation that kept them from having to write all three equations, keeping their focus on the form of the equation they sought. This created the same result as Solution 1, $y=-\frac{1}{3}x^2+\frac{1}{3}x+2$. CAS Aside: No students offered these next two solutions, but I believe when using a CAS, it is important for users to remember that the machine typically does not care what output form you want. The standard form is the only “algebraically simple” approach when setting up a solution by hand, but the availability of technology makes solving for any form equally accessible. The next screen shows that the vertex and factored forms are just as easily derived as the standard form my students found in Solution 2. I was surprised when the last line’s output wasn’t in vertex form, $y=-\frac{1}{3}\cdot \left ( x-\frac{1}{2} \right )^2+\frac{25}{12}$, but the coefficients in its expanded form clearly show the equivalence between this form and the standard forms derived in Solutions 1 and 2–a valuable connection. Solution 3–Symmetry: Two students said they noticed that $f(0)=f(1)=2$ guaranteed the vertex of the parabola occurred at $x=\frac{1}{2}$. Because $f(3)=0$ defined one real root of the unknown quadratic, the parabola’s symmetry guaranteed another at $x=-2$, giving potential equation $y=a\cdot (x-3)(x+2)$. They substituted the given (0,2) to solve for a, giving final equation $y=-\frac{1}{3}\cdot (x-3)(x+2)$ as confirmed by the CAS approach above.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
Solution 4–Transformations: One of the big lessons I repeat in every class I teach is this: If you don’t like how a question is posed. Change it! Notice that two of the given points have the same y-coordinate. If that y-coordinate had been 0 (instead of its given value, 2), a factored form would be simple. Well, why not force them to be x-intercepts by translating all of the given points down 2 units? The transformed data show x-intercepts at 0 and 1 with another ordered pair at $(3,-2)$. From here, the factored form is easy: $y=a\cdot (x-0)(x-1)$. Substituting $(3,-2)$ gives $a=-\frac{1}{3}$ and the final equation is $y=-\frac{1}{3}\cdot (x-0)(x-1)$ . Of course, this is an equation for the transformed points. Sliding the result back up two units, $y=-\frac{1}{3}\cdot (x-0)(x-1)+2$, gives an equation for the given points. Aside from its lead coefficient, this last equation looked very different from the other forms, but some quick expansion proved its equivalence. Conclusion: It would have been nice if someone had used the symmetry noted in Solution 3 to attempt a vertex-form answer via systems. Given the vertex at $x=\frac{1}{2}$ with an unknown y-coordinate, a potential equation is $y=a\cdot \left ( x-\frac{1}{2} \right )^2+k$. Substituting $(3,0)$ and either $(0,2)\text{ or }(1,2)$ creates a 2×2 system of linear equations, $\left\{\begin{matrix} 0=a\cdot \left ( 3-\frac{1}{2} \right )^2+k \\ 2=a\cdot \left ( 0-\frac{1}{2} \right )^2+k \end{matrix}\right.$. From there, a by-hand or CAS solution would have been equally acceptable to me. That the few alternative approaches I offered above weren’t used didn’t matter in the end. My students were creative, followed their own instincts to find solutions that aligned with their thinking, and clearly appreciated the alternative ways their classmates used to find answers. Creativity and individual expression reigned, while everyone broadened their understanding that there’s not just one way to do math.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
It was a good day. ## Cubics and CAS Here’s a question I posed to one of my precalculus classes for homework at the end of last week along with three solutions we developed. Suppose the graph of a cubic function has an inflection point at (1,3) and passes through (0,-4). 1. Name one other point that MUST be on the curve, and 2. give TWO different cubic equations that would pass through the three points. SOLUTION ALERT! Don’t read any further if you want to solve this problem for yourself. The first question relies on the fact that every cubic function has 180 degree rotational symmetry about its inflection point. This is equivalent to saying that the graph of a cubic function is its own image when the function’s graph is reflected through its inflection point. That means the third point is the image of (0,-4) when point-reflected through the inflection point (1,3), which is the point (2,10) as shown graphically below. From here, my students came up with 2 different solutions to the second question and upon prodding, we created a third. SOLUTION 1: Virtually every student assumed $y=a\cdot x^3$ was the parent function of a cubic with unknown leading coefficient whose “center” (inflection point) had been slid to (1,3). Plugging in the given (0,-4) to $(y-3)=a\cdot (x-1)^3$ gives $a=7$. Here’s their graph.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
SOLUTION 2: Many students had difficulty coming up with another equation. A few could sketch in other cubic graphs (curiously, all had positive lead coefficients) that contained the 3 points above, but didn’t know how to find equations. That’s when Sara pointed out that if the generic expanded form of a cubic was $a\cdot x^3+b\cdot x^2 +c\cdot x+d$ , then any 4 ordered pairs with unique x-coordinates should define a unique cubic. That is, if we picked any 4th point with x not 0, 1, or 2, then we should get an equation. That this would create a 4×4 system of equations didn’t bother her at all. She knew in theory how to solve such a thing, but she was thinking on a much higher plane. Her CAS technology expeditiously did the grunt work, allowing her brain to keep moving. A doubtful classmate called out, “OK. Make it go through (100,100).” Following is a CAS screen roughly duplicating Sara’s approach and a graph confirming the fit. The equation was onerous, but with a quick copy-paste, Sara had moved from idea to computation to ugly equation and graph in just a couple minutes. The doubter was convinced and the “wow”s from throughout the room conveyed the respect for the power of a properly wielded CAS. In particular, notice how the TI-Nspire CAS syntax in lines 1 and 3 keep the user’s focus on the type of equation being solved and eliminates the need to actually enter 4 separate equations. It doesn’t always work, but it’s a particularly lovely piece of scaffolding when it does. SOLUTION 3: One of my goals in Algebra II and Precalculus courses is to teach my students that they don’t need to always accept problems as stated. Sometimes they can change initial conditions to create a much cleaner work environment so long as they transform their “clean” solution back to the state of the initial conditions.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
In this case, I asked what would happen if they translated the inflection point using $T_{-1,-3}$ to the origin, making the other given point (-1,-7). Several immediately called the 3rd point to be (1,7) which “untranslating” — $T_{1,3}(1,7)=(2,10)$ — confirmed to be the earlier finding. For cases where the cubic had another real root at $x=r$, then symmetry immediately made $x=-r$ another root, and a factored form of the equation becomes $y=a\cdot (x)(x-r)(x+r)$ for some value of a. Plugging in (-1,-7) gives a in terms of r. The last line slid the initially translated equation using $T_{1,3}$ to re-position the previous line according to the initial conditions. While unasked for, notice how the CAS performed some polynomial division on the right-side expression. I created a GeoGebra document with a slider for the root using the equation from the last line of the CAS image above. The image below shows one possible position of the retranslated root. If you want to play with a live version of this, you will need a free copy of GeoGebra to run it, but the file is here. What’s nice here is how the problem became one of simple factors once the inflection point was translated to the origin. Notice also that the CAS version of the equation concludes with $+7x-4$, the line containing the original three points. This is nice for two reasons. The numerator of the rational term is $-7x(x-2)(x-1)$ which zeros out the fraction at x=0, 1, or 2, putting the cubic exactly on the line $y=7x-4$ at those points. The only r-values are in the denominator, so as $r\rightarrow\infty$, the rational term also becomes zero. Graphically, you can see this happen as the cubic “unrolls” onto $y=7x-4$ as you drag $\|x\|\rightarrow\infty$. Essentially, this shows both graphically and algebraically that $y=7x-4$ is the limiting degenerate curve the cubic function approaches as two of its transformed real roots grow without bound. ## Recognizing Patterns	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
## Recognizing Patterns I’ve often told my students that the best problem solvers are those who recognize patterns from past problems in new situations. So, the best way to become a better problem solver is to solve lots of problems, study the ways others have solved the problems you’ve already cracked (or at least attempted), and to keep pushing your boundaries because you never know what parts of what you learn may end up providing unexpected future insights. I lay no claims to being a great problem solver, but I absolutely benefited from problem solving exposure when I encountered @jamestanton‘s latest “Playing with Numbers” puzzler from his May 2012 Cool Math Newsletter. (Click here to access all of Jim’s newsletters). (BTW, Jim’s Web page is chock full of amazing videos and insights both on the problem-solving front and for those interested in curriculum discussions.) Here’s what Jim posed: Write the numbers 1 though 10 on the board. Pick any two numbers, erase them, and replace them with the single number given by their sum plus their product. (So, if you choose to erase the numbers a and b, replace them with a + b + ab .) You now have nine numbers on the board. Do this again: Pick any two numbers, erase them, and replace them with their sum plus product. You now have eight numbers on the board. Do this seven more times until you have a single number on the board. Why do all who play this game end up with the same single number at the end? What is that final number? SOLUTION ALERT: Don’t read any further if you want to solve this yourself.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
SOLUTION ALERT: Don’t read any further if you want to solve this yourself. I started small and general. If the first two numbers (a and b) produce $a+b+ab$, then adding c to the original list gives $[(a+b+ab)+c]+ [(a+b+ab)\cdot c]$ which can be rewritten as $a+b+c+ab+ac+bc+abc$. That’s when the intuition struck. Notice that the rewritten form is the sum of every individual number in the list, AND every possible pair of those numbers, AND concludes with the product of the three numbers. I’ve seen that pattern before! Given an nth degree polynomial with roots $r_1, r_2, \ldots r_n$, it’s factored form is $(x-r_1)\cdot (x-r_2)\cdot\ldots\cdot (x-r_n)$ which can be expanded and rewritten as $x^n-(r_1+r_2+\ldots+r_n)\cdot x^{n-1}+$ $+(\text{every pair-wise product of } r_1 \ldots r_n)\cdot x^{n-2}-$ $-(\text{every three-way product of } r_1 \ldots r_n)\cdot x^{n-3}+\ldots$ $\ldots \pm (r_1\cdot r_2\cdot \ldots\cdot r_n)$ Where the sign of the final term is positive for even n and negative for odd n. I saw this in many algebra textbooks when I started teaching over 20 years ago, but haven’t encountered it lately. Then again, I haven’t been looking for it. Here’s the point … other than the alternating signs, the coefficients of the expanded polynomial are exactly identical to the sums I was getting from Jim’s problem. That’s when I rewrote my original problem on my CAS to $(x+a)\cdot (x+b)\cdot (x+c)$ and expanded it (see line 1 below). The coefficients of $x^2$ are the individual numbers, the coefficients of $x$ are all the pair-wise combinations, and the constant term drifting off the end of the line is $a\cdot b\cdot c$. And I eliminated the $\pm$ sign challenge by individually adding the numbers instead of subtracting them as I had in the polynomial root example that inspired my insight. Let $x=1$ to clean up and make the pattern more obvious.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
So, creating a polynomial with the given numbers and substituting $x=1$ will create a number one more than the sum I wanted (note the extra +1 resulting from the coefficient of $x^n$). Using pi notation, substituting $x=1$, and subtracting 1 gives the answer. In case you didn’t know, pi notation works exactly the same as sigma notation, but you multiply the terms instead of adding them. The solution–39,916,799–is surprisingly large given the initial problem statement, and while my CAS use confirmed my intuition and effortlessly crunched the numbers, its tendency to multiply numbers whenever encountered has actually hidden something pretty. From the top line of the last image, substituting $x=1$ would have created the product $2\cdot 3\cdot 4\cdot\ldots \cdot 11$ which the penultimate line computed to be 39916800. But before the product, that number was $11!$, making $11!-1$ a far more revealing version of the solution. MORAL: Even after you have an answer, take some time to review what has happened to give yourself a chance to learn even more. EXTENSION 1: Instead of 1 to 10 as the initial numbers, what if the list went from 1 to any positive integer n? Prove that the final number on the board is $(n+1)!-1$. EXTENSION 2: While a positive integer sequence starting at 1 (or 0) produces a nice factorial in the answer, this approach can be used with any number list. For example, follow Jim’s rules with 37, 5, -2, and 7.9. Use the polynomial approach below for a quick solution of -2030.2. Confirm using the original rules if you need. EXTENSION 3: By now it should be obvious that any list of numbers can be used in this problem. Prove that every list of numbers which includes -1 has the same solution.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
EXTENSION 4: Before explaining some lovely extensions of problems like this to generalized commutativity and associativity, Jim’s May 12 Cool Math Newsletter asks what would happen if instead of $a+b+ab$, the rule for combining a and b was $\displaystyle\frac{a\cdot b}{a+b}$. You can show that with three terms, this would become $\displaystyle\frac{abc}{ab+ac+bc}$, and four terms would give $\displaystyle\frac{abcd}{abc+abd+acd+bcd}$. In other words, this rule would morph n original numbers into a fraction whose numerator is the product of the numbers and whose denominator is the sum of all possible products of any (n-1)-sized groups of those numbers. For the original integers 1 to 10, I know the numerator and denominator terms are the last two coefficients in a polynomial expansion. The final fraction simplifies, but I think $\displaystyle\frac{3628800}{10628640}$ is slightly more informative. Happy thoughts, problems, solutions, and connections to you all.	{ "domain": "wordpress.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9907319878505035, "lm_q1q2_score": 0.8962662254973889, "lm_q2_score": 0.9046505376715775, "openwebmath_perplexity": 980.6773426566747, "openwebmath_score": 0.803494930267334, "tags": null, "url": "https://casmusings.wordpress.com/tag/polynomial/" }
GMAT Question of the Day: Daily via email \| Daily via Instagram New to GMAT Club? Watch this Video It is currently 22 Feb 2020, 18:16 GMAT Club Daily Prep Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History What is the greatest value of y such that 4^y is a factor of 9! ? Author Message TAGS: Hide Tags Math Expert Joined: 02 Sep 2009 Posts: 61396 What is the greatest value of y such that 4^y is a factor of 9! ? [#permalink] Show Tags 08 Dec 2016, 11:57 00:00 Difficulty: 15% (low) Question Stats: 77% (01:07) correct 23% (01:20) wrong based on 60 sessions HideShow timer Statistics What is the greatest value of y such that 4^y is a factor of 9! ? A. 5 B. 4 C. 3 D. 1 E. 0 _________________ Director Joined: 05 Mar 2015 Posts: 960 What is the greatest value of y such that 4^y is a factor of 9! ? [#permalink] Show Tags 08 Dec 2016, 19:02 1 Bunuel wrote: What is the greatest value of y such that 4^y is a factor of 9! ? A. 5 B. 4 C. 3 D. 1 E. 0 4^y=2^(2y) no. of 2's in 9! 9/2=4 9/2^2=2 9/2^3=1 total= 4+2+1=7 so as 2y=7 we get y=3 Ans C Manager Joined: 27 Aug 2015 Posts: 86 Re: What is the greatest value of y such that 4^y is a factor of 9! ? [#permalink] Show Tags 09 Dec 2016, 02:25 1 The formula for such problems is like 9 /4= 2 9/4^2=0 Total = 2 However answer should be 3 if we actually count it. Where am I going wrong? Posted from my mobile device Board of Directors Status: QA & VA Forum Moderator Joined: 11 Jun 2011 Posts: 4841 Location: India GPA: 3.5 Re: What is the greatest value of y such that 4^y is a factor of 9! ? [#permalink] Show Tags	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 1, "lm_q1q2_score": 0.8962513786759491, "lm_q2_score": 0.8962513786759491, "openwebmath_perplexity": 2563.270617579204, "openwebmath_score": 0.7827305197715759, "tags": null, "url": "https://gmatclub.com/forum/what-is-the-greatest-value-of-y-such-that-4-y-is-a-factor-of-230289.html" }
Show Tags 09 Dec 2016, 10:18 2 Bunuel wrote: What is the greatest value of y such that 4^y is a factor of 9! ? A. 5 B. 4 C. 3 D. 1 E. 0 $$9! = 987654321$$ Or, $$9! = 3^22^372352^2321$$ Or, $$9! = 2^73^457$$ Now, $$2^7 = 4^32$$ Thus, we have the greatest value of y = 3 , hence answer will be (C) rakaisraka hope its clear with you ... Further I suggest you go through the concept once again to clear your doubts here math-number-theory-88376.html#p666609 _________________ Thanks and Regards Abhishek.... PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS How to use Search Function in GMAT Club \| Rules for Posting in QA forum \| Writing Mathematical Formulas \|Rules for Posting in VA forum \| Request Expert's Reply ( VA Forum Only ) Director Joined: 05 Mar 2015 Posts: 960 Re: What is the greatest value of y such that 4^y is a factor of 9! ? [#permalink] Show Tags 09 Dec 2016, 10:35 1 rakaisraka wrote: The formula for such problems is like 9 /4= 2 9/4^2=0 Total = 2 However answer should be 3 if we actually count it. Where am I going wrong? Posted from my mobile device rakaisraka when u r finding 4^y means u have to count every 2's .. suppose if it was 10! then it must have 12...6...10 then it has 6=23 && 10=25 where one no. 2 from 6 and one no. 2 from 10 also counted as a 4 in 10! let me make more clear if u have to find 6^y in X! as 6=23 then u have to count every 2 and every 3 in X! and the minimum pair of 2&3 will make the answer hope it is clear Target Test Prep Representative Affiliations: Target Test Prep Joined: 04 Mar 2011 Posts: 2801 Re: What is the greatest value of y such that 4^y is a factor of 9! ? [#permalink] Show Tags 12 Dec 2016, 17:16 1 Bunuel wrote: What is the greatest value of y such that 4^y is a factor of 9! ? A. 5 B. 4 C. 3 D. 1 E. 0 Since 4 = 2^2, we are actually trying to determine the largest value y such that 2^(2y) is a factor of 9!.	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 1, "lm_q1q2_score": 0.8962513786759491, "lm_q2_score": 0.8962513786759491, "openwebmath_perplexity": 2563.270617579204, "openwebmath_score": 0.7827305197715759, "tags": null, "url": "https://gmatclub.com/forum/what-is-the-greatest-value-of-y-such-that-4-y-is-a-factor-of-230289.html" }
Let’s first determine the number of factors of 2 within 9!. To do that, we can use the following shortcut in which we divide 9 by 2, and then divide the quotient of 9/2 by 2 and continue this process until we can no longer get a nonzero integer as the quotient. 9/2 = 4 (we can ignore the remainder) 4/2 = 2 2/2 = 1 Since 1/2 does not produce a nonzero quotient, we can stop. The final step is to add up our quotients; that sum represents the number of factors of 2 within 9!. Thus, there are 4 + 2 + 1 = 7 factors of 2 within 9! However, we are not asked for the number of factors of 2; instead we are asked for the number of factors of 4. We see that 7 factors of 2 will produce 3 factors of 4. _________________ Jeffrey Miller Jeff@TargetTestPrep.com 181 Reviews 5-star rated online GMAT quant self study course See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Non-Human User Joined: 09 Sep 2013 Posts: 14124 Re: What is the greatest value of y such that 4^y is a factor of 9! ? [#permalink] Show Tags 17 Jan 2020, 02:54 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Re: What is the greatest value of y such that 4^y is a factor of 9! ? [#permalink] 17 Jan 2020, 02:54 Display posts from previous: Sort by	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 1, "lm_q1q2_score": 0.8962513786759491, "lm_q2_score": 0.8962513786759491, "openwebmath_perplexity": 2563.270617579204, "openwebmath_score": 0.7827305197715759, "tags": null, "url": "https://gmatclub.com/forum/what-is-the-greatest-value-of-y-such-that-4-y-is-a-factor-of-230289.html" }
# On Recurring decimals $\large{0\red{.}\overline{x_1x_2x_3...x_n}=\dfrac{x_1x_2x_3...x_n}{10^n-1}}$ Proof of the above statement Let $l=0\red{.}\overline{x_1x_2x_3...x_n}$ $10^nl=x_1x_2x_3...x_n\red{.}\overline{x_1x_2x_3...x_n}$ $\Rightarrow 10^nl-l={x_1x_2x_3...x_n}$ $(10^n-1)l=x_1x_2x_3...x_n$ ${l=\dfrac{x_1x_2x_3...x_n}{10^n-1}}$ $\boxed{0\red{.}\overline{x_1x_2x_3...x_n}=\dfrac{x_1x_2x_3...x_n}{10^n-1}}$ $\large{a_1a_2a_3...a_p\red{.}b_1b_2b_3...b_q\overline{x_1x_2x_3...x_n}=\dfrac{1}{10^q}(10^q\times a_1a_2a_3...a_p+b_1b_2b_3...b_q+\dfrac{x_1x_2x_3...x_n}{10^n-1})}$ Proof of the above statement For any number $a_1a_2a_3...a_p\red{.}b_1b_2b_3...b_q\overline{x_1x_2x_3...x_n}$ $a_1a_2a_3...a_p\red{.}b_1b_2b_3...b_q\overline{x_1x_2x_3...x_n}=a_1a_2a_3...a_p+0\red{.}b_1b_2b_3...b_q\overline{x_1x_2x_3...x_n}$ $=\dfrac{1}{10^q}(10^q\times a_1a_2a_3...a_p+b_1b_2b_3...b_q\red{.}\overline{x_1x_2x_3...x_n})$ $=\dfrac{1}{10^q}(10^q\times a_1a_2a_3...a_p+b_1b_2b_3...b_q+0\red{.}\overline{x_1x_2x_3...x_n})$ $=\boxed{\dfrac{1}{10^q}(10^q\times a_1a_2a_3...a_p+b_1b_2b_3...b_q+\dfrac{x_1x_2x_3...x_n}{10^n-1})}$ Note : • $x_1x_2$ act as number with digits $x_1,x_2$ for example if $x_1=5$ and $x_2=8\Rightarrow x_1x_2=58$ dont confuse ($x_1x_2\cancel{=}x_1\times x_2$), same for $x_1x_2x_3$ and $x_1x_2x_3...x_{n-1}x_n$ • $0\red{.}\overline{a}=0\red{.}aaaaa...$ Note by Zakir Husain 6 months, 2 weeks ago This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science. When posting on Brilliant:	{ "domain": "brilliant.org", "id": null, "lm_label": "1. Yes\n2. Yes\n\n", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9664104885453715, "lm_q1q2_score": 0.8962136073596833, "lm_q2_score": 0.9273632871148287, "openwebmath_perplexity": 2703.0319096444596, "openwebmath_score": 0.9654994010925293, "tags": null, "url": "https://brilliant.org/discussions/thread/on-recurring-decimals/" }
When posting on Brilliant: • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused . • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone. • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge. • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events. MarkdownAppears as italics or _italics_ italics bold or __bold__ bold - bulleted- list • bulleted • list 1. numbered2. list 1. numbered 2. list Note: you must add a full line of space before and after lists for them to show up correctly paragraph 1paragraph 2 paragraph 1 paragraph 2 [example link](https://brilliant.org)example link > This is a quote This is a quote # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" # I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world" MathAppears as Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting. 2 \times 3 $2 \times 3$ 2^{34} $2^{34}$ a_{i-1} $a_{i-1}$ \frac{2}{3} $\frac{2}{3}$ \sqrt{2} $\sqrt{2}$ \sum_{i=1}^3 $\sum_{i=1}^3$ \sin \theta $\sin \theta$ \boxed{123} $\boxed{123}$ Sort by: - 6 months, 2 weeks ago Well.. I am impressed. - 6 months, 2 weeks ago ✨ brilliant +1 - 6 months, 2 weeks ago By the way, I have an interesting #Geometry problem! Given points $A,B,C,D$, find the square $\square PQRS$ with A on PQ, B on QR, C on RS, D on SP. I figured out the first part, where we can construct circles with diameters AB, BC, CD, DA respectively, so if a point W is on arc AB, $\angle AWB=90^\circ.$ - 6 months, 2 weeks ago	{ "domain": "brilliant.org", "id": null, "lm_label": "1. Yes\n2. Yes\n\n", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9664104885453715, "lm_q1q2_score": 0.8962136073596833, "lm_q2_score": 0.9273632871148287, "openwebmath_perplexity": 2703.0319096444596, "openwebmath_score": 0.9654994010925293, "tags": null, "url": "https://brilliant.org/discussions/thread/on-recurring-decimals/" }
- 6 months, 2 weeks ago I tried the problem and got an algorithm to construct a rectangle $PQRS$ with points $A,B,C$ and $D$ on sides $PQ,QR,RS,SP$ respectively. Also there will be infinitely many such rectangles for given points $A,B,C,D$ - 6 months, 2 weeks ago What about a square? - 6 months, 2 weeks ago I will try it also! and will inform you as I get any results. - 6 months, 2 weeks ago Let’s start a discussion! That might help :) - 6 months, 2 weeks ago square is also a rectangle.. - 6 months, 2 weeks ago But a rectangle isn’t a square, so I hope to find an algorithm to construct a square (I know it is possible but I don’t know a specific way to do it except for brute-force :P) :) - 6 months, 2 weeks ago	{ "domain": "brilliant.org", "id": null, "lm_label": "1. Yes\n2. Yes\n\n", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9664104885453715, "lm_q1q2_score": 0.8962136073596833, "lm_q2_score": 0.9273632871148287, "openwebmath_perplexity": 2703.0319096444596, "openwebmath_score": 0.9654994010925293, "tags": null, "url": "https://brilliant.org/discussions/thread/on-recurring-decimals/" }
# greater than or equal to sign	{ "domain": "zong-music.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9621075722839015, "lm_q1q2_score": 0.896167132625336, "lm_q2_score": 0.9314625083949473, "openwebmath_perplexity": 948.9165942980742, "openwebmath_score": 0.5518640279769897, "tags": null, "url": "https://zong-music.com/ha2ten/greater-than-or-equal-to-sign-9b504e" }
For example, x ≥ -3 is the solution of a certain expression in variable x. Select Symbol and then More Symbols. For example, the symbol is used below to express the less-than-or-equal relationship between two variables: ≥. "Greater than or equal to", as the suggests, means something is either greater than or equal to another thing. is less than > > is greater than ≮ \nless: is not less than ≯ \ngtr: is not greater than ≤ \leq: is less than or equal to ≥ \geq: is greater than or equal to ⩽ \leqslant: is less than or equal to ⩾ 923 Views. Use the appropriate math symbol to indicate "greater than", "less than" or "equal to" for each of the following: a. Greater than or equal application to numbers: Syntax of Greater than or Equal is A>=B, where A and B are numeric or Text values. With Microsoft Word, inserting a greater than or equal to sign into your Word document can be as simple as pressing the Equal keyboard key or the Greater Than keyboard key, but there is also a way to insert these characters as actual equations. For example, 4 or 3 ≥ 1 shows us a greater sign over half an equal sign, meaning that 4 or 3 are greater than or equal to 1. In such cases, we can use the greater than or equal to symbol, i.e. In Greater than or equal operator A value compares with B value it will return true in two cases one is when A greater than B and another is when A equal to B. Rate this symbol: (3.80 / 5 votes) Specifies that one value is greater than, or equal to, another value. This symbol is nothing but the "greater than" symbol with a sleeping line under it. Less Than or Equal To (<=) Operator. “Greater than or equal to” and “less than or equal to” are just the applicable symbol with half an equal sign under it. Greater Than or Equal To: Math Definition. 2 ≥ 2. But, when we say 'at least', we mean 'greater than or equal to'. The less than or equal to symbol is used to express the relationship between two quantities or as a boolean logical operator. "Greater than or equal to"	{ "domain": "zong-music.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9621075722839015, "lm_q1q2_score": 0.896167132625336, "lm_q2_score": 0.9314625083949473, "openwebmath_perplexity": 948.9165942980742, "openwebmath_score": 0.5518640279769897, "tags": null, "url": "https://zong-music.com/ha2ten/greater-than-or-equal-to-sign-9b504e" }
the relationship between two quantities or as a boolean logical operator. "Greater than or equal to" is represented by the symbol " ≥ ≥ ". Solution for 1. The greater-than sign is a mathematical symbol that denotes an inequality between two values. In an acidic solution [H]… Greater than or Equal in Excel – Example #5. Here a could be greater … Examples: 5 ≥ 4. The sql Greater Than or Equal To operator is used to check whether the left-hand operator is higher than or equal to the right-hand operator or not. Category: Mathematical Symbols. When we say 'as many as' or 'no more than', we mean 'less than or equal to' which means that a could be less than b or equal to b. Select the Greater-than Or Equal To tab in the Symbol window. use ">=" for greater than or equal use "<=" for less than or equal In general, Sheets uses the same "language" as Excel, so you can look up Excel tips for Sheets. Copy the Greater-than Or Equal To in the above table (it can be automatically copied with a mouse click) and paste it in word, Or. Finding specific symbols in countless symbols is obviously a waste of time, and some characters like emoji usually can't be found. Graphical characteristics: Asymmetric, Open shape, Monochrome, Contains straight lines, Has no crossing lines. Select the Insert tab. If left-hand operator higher than or equal to right-hand operator then condition will be true and it will return matched records. Sometimes we may observe scenarios where the result obtained by solving an expression for a variable, which are greater than or equal to each other. As we saw earlier, the greater than and less than symbols can also be combined with the equal sign. Is obviously a waste of time, and some characters like emoji usually ca n't be found ≥ ... Mean 'greater than or equal to '' is represented by the symbol.! Under it we say 'at least ', we mean 'greater than or equal to another thing return matched.. Expression in variable x is represented by the symbol ≥ ≥ nothing but the	{ "domain": "zong-music.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9621075722839015, "lm_q1q2_score": 0.896167132625336, "lm_q2_score": 0.9314625083949473, "openwebmath_perplexity": 948.9165942980742, "openwebmath_score": 0.5518640279769897, "tags": null, "url": "https://zong-music.com/ha2ten/greater-than-or-equal-to-sign-9b504e" }
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# For $t\in [ 0, 1 )$ is $\frac{xe^{tx}}{e^{x}-1}$ integrable over $x\in (0 , \infty )$? For $t\in [ 0, 1 )$ is $$\frac{xe^{tx}}{e^{x}-1}$$ integrable over $x\in (0 , \infty )$? I.e., $$\int_{0}^{\infty} \frac{xe^{tx}}{e^{x}-1} dx < \infty?$$ How do I show this? - As $x\to0$, $x/(e^x-1)$ approaches a finite limit. As $x\to\infty$, do a limit-comparison of the integrand to $xe^{tx}/e^x$. - As $\frac x{e^x-1}$ as a limit when $x\to 0$ (namely $1$), the only problem is when $x\to\infty$. We have $e^x-1\sim e^x$ at $+\infty$, so $\dfrac{xe^{tx}}{e^x-1}\sim xe^{(t-1)x}$. Using Taylor's series, $$e^{(t-1)x}\leq \frac 1{1+(1-t)x+x^2(1-t)^2/2+x^3(1-t)^3/6},$$ the integral is convergent for $t\in[0,1)$. - Doesn't the first limit go to $1$ instead of $e^{–1}$? – Pedro Tamaroff Oct 23 '12 at 13:05 @PeterTamaroff Right. Fixed now. – Davide Giraudo Oct 23 '12 at 13:14 What matters in the improper integral of a nice function (e.g. elementary function) is the existence of singularities. In a broad sense, there are two kinds of singularities that counts. 1. A point where the integrand does not behave well. For example, the function can explode to infinite or oscillate infinitely. 2. A point at infinity. That is, $\pm \infty$. Away from singularities, the behavior of the function is quite under control, allowing us to concentrate our attention on those singularities. There is a basic method to establish the convergence (or divergence) of the integral near each singularity point. In many cases, except for the oscillatory case, you can find a dominating function that determines the order of magnitude of the function near the point. If the dominating function is easy to integrate, then you can make a comparison with this dominating function to conclude the convergence behavior. For example, let us consider $$\int_{0}^{\frac{\pi}{2}} \tan^2 x \, dx \quad \text{and} \quad \int_{0}^{\infty} \frac{x^2 e^{-x}}{1+x^2} \, dx.$$	{ "domain": "stackexchange.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9732407160384083, "lm_q1q2_score": 0.8961499950229149, "lm_q2_score": 0.9207896671963207, "openwebmath_perplexity": 161.082132305399, "openwebmath_score": 0.9804616570472717, "tags": null, "url": "http://math.stackexchange.com/questions/219329/for-t-in-0-1-is-fracxetxex-1-integrable-over-x-in-0-in/219347" }
We can easily check that $\tan^2 x$ is bounded below by $(x-\frac{\pi}{2})^{-2}$ near the singularity $x = \frac{\pi}{2}$ and $x^2 e^{-x} / (1 + x^2)$ is bounded above by $e^{-x}$ near the singularity $x = \infty$. Then $$\int_{\frac{\pi}{2}-\delta}^{\frac{\pi}{2}} \tan^2 x \, dx \geq \int_{\frac{\pi}{2}-\delta}^{\frac{\pi}{2}} \left(x - \frac{\pi}{2}\right)^{2} \, dx = \infty$$ for sufficiently small $\delta > 0$ and $$\int_{R}^{\infty} \frac{x^2 e^{-x}}{1+x^2}\,dx \leq \int_{R}^{\infty} e^{-x} \, dx < \infty$$ for sufficiently large $R > 0$. Thus we find that the former diverges to $\infty$ and the latter converges. In our example, there are two seemingly singular points, namely $x = 0$ and $x = \infty$. At $x = 0$, we find that $$\lim_{x \to 0} \frac{x e^{tx}}{e^x - 1} = 1.$$ This means that this singularity is removable, in the sense that the function can be extended in a continuous manner to this point. Thus we need not count this point and we can move our attention to the point at infinity. To establish the convergence (or possibly divergence) of the integral near $x = \infty$, we write $$\frac{x e^{tx}}{e^x - 1} = \frac{x}{1 - e^{-x}} e^{-(1-t)x}.$$ It is clear that for sufficiently large $x$, the term $\frac{x}{1 - e^{-x}}$ is bounded above by some constant $C > 0$. Thus the dominating function is $e^{-(1-t)x}$ and $$\int_{R}^{\infty} \frac{x e^{tx}}{e^x - 1} \, dx \leq \int_{R}^{\infty} C e^{-(1-t)x} \, dx < \infty$$ for large $R$. Therefore the improper integral converges. - Thanks for your answers and especially for this limit method. But in this way, I actually found a simpler bound, namely, the following: observe $$\frac{xe^{tx}}{e^{x}-1}=\frac{xe^{(1/2)(t-1)x}}{1-e^{-x}}e^{(1/2)(t-1)x}$$ $$\frac{xe^{(1/2)(t-1)x}}{1-e^{-x}}<M$$ for a constant $M>0$ as $$\frac{xe^{(1/2)(t-1)x}}{1-e^{-x}}$$ is continuous and the limits for $x\to 0$ and $x \to \infty$ are finite. And this can be directly used for the integrability of the function. -	{ "domain": "stackexchange.com", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9732407160384083, "lm_q1q2_score": 0.8961499950229149, "lm_q2_score": 0.9207896671963207, "openwebmath_perplexity": 161.082132305399, "openwebmath_score": 0.9804616570472717, "tags": null, "url": "http://math.stackexchange.com/questions/219329/for-t-in-0-1-is-fracxetxex-1-integrable-over-x-in-0-in/219347" }
GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 17 Jun 2019, 08:03 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # The price of a consumer good increased by p%. . . Author Message TAGS: ### Hide Tags e-GMAT Representative Joined: 04 Jan 2015 Posts: 2888 The price of a consumer good increased by p%. . . [#permalink] ### Show Tags Updated on: 07 Aug 2018, 07:09 1 12 00:00 Difficulty: 55% (hard) Question Stats: 68% (02:30) correct 32% (02:34) wrong based on 273 sessions ### HideShow timer Statistics The price of a consumer good increased by $$p$$% during $$2012$$ and decreased by $$12$$% during $$2013$$. If no other change took place in the price of the good and the price of the good at the end of $$2013$$ was $$10$$% higher than the price at the beginning of $$2012$$, what was the value of $$p$$? A. $$-2$$% B. $$2$$% C. $$22$$% D. $$25$$% E. Cannot be determined Take a stab at this fresh question from e-GMAT. Post your analysis below. Official Solution to be provided after receiving some good analyses. _________________ Originally posted by EgmatQuantExpert on 11 Nov 2016, 05:46. Last edited by EgmatQuantExpert on 07 Aug 2018, 07:09, edited 1 time in total. CEO Joined: 12 Sep 2015 Posts: 3777 Re: The price of a consumer good increased by p%. . . [#permalink] ### Show Tags	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
### Show Tags 11 Nov 2016, 07:09 Top Contributor EgmatQuantExpert wrote: The price of a consumer good increased by p% during 2012 and decreased by 12% during 2013. If no other change took place in the price of the good and the price of the good at the end of 2013 was 10% higher than the price at the beginning of 2012, what was the value of p? A. $$-2$$% B. $$2$$% C. $$22$$% D. $$25$$% E. Cannot be determined Let $100 be the original price The price of a consumer good increased by p% during 2012 p% = p/100, so a p% INCREASE is the same a multiplying the original price by 1 + p/100 So, the new price = ($100)(1 + p/100)	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
The price then decreased by 12% during 2013 A 12% DECREASE is the same a multiplying the price by 0.88 So, the new price = ($100)(1 + p/100)(0.88) The price of the good at the end of 2013 was 10% higher than the price at the beginning of 2012 If the original price was$100, then the price at the end of 2013 was $110 So, we can write:$110 = ($100)(1 + p/100)(0.88) Simplify:$110 = (100 + p)(0.88)	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
Simplify more: $110 = 88 + 0.88p Subtract 88 from both sides: 22 = 0.88p So, p = 22/0.88 = 25 Answer: RELATED VIDEO _________________ Test confidently with gmatprepnow.com Board of Directors Status: QA & VA Forum Moderator Joined: 11 Jun 2011 Posts: 4499 Location: India GPA: 3.5 WE: Business Development (Commercial Banking) Re: The price of a consumer good increased by p%. . . [#permalink] ### Show Tags 11 Nov 2016, 12:48 EgmatQuantExpert wrote: The price of a consumer good increased by $$p$$% during $$2012$$ and decreased by $$12$$% during $$2013$$. If no other change took place in the price of the good and the price of the good at the end of $$2013$$ was $$10$$% higher than the price at the beginning of $$2012$$, what was the value of $$p$$? A. $$-2$$% B. $$2$$% C. $$22$$% D. $$25$$% E. Cannot be determined Price's corresponding to year - 2011 = $$100$$ 2012 = $$100 + p$$ 2013 = $$\frac{88}{100}(100 + p)$$ Further , $$\frac{88}{100}(100 + p)$$ = $$110$$ Or, $$\frac{8}{100}(100 + p)$$ = $$10$$ Or, 800 + 8p = 1000 Or, 8p = 200 So, p = 25% Hence, answer will be (D) 25% _________________ Thanks and Regards Abhishek.... PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS How to use Search Function in GMAT Club \| Rules for Posting in QA forum \| Writing Mathematical Formulas \|Rules for Posting in VA forum \| Request Expert's Reply ( VA Forum Only ) Current Student Joined: 26 Jan 2016 Posts: 100 Location: United States GPA: 3.37 Re: The price of a consumer good increased by p%. . . [#permalink] ### Show Tags 11 Nov 2016, 13:01 Lets start with a number for the original value. 100 is the easiest. So we're looking for a value of 110 at the end of 2013. Just by looking at the values we can get an idea of what to start testing. If we're increasing 100 by p% then decreasing it by 12% and the original value is still 10% higher we need a value much higher than 12. 25% is the easiest value to start with. 100+25%=125	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
10% higher we need a value much higher than 12. 25% is the easiest value to start with. 100+25%=125 125-12%=110 D Current Student Joined: 12 Aug 2015 Posts: 2610 Schools: Boston U '20 (M) GRE 1: Q169 V154 Re: The price of a consumer good increased by p%. . . [#permalink] ### Show Tags 12 Nov 2016, 20:32 For all the algebra loving people out there=> Let price at the beginning of 2012 be$x	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
so after the end of 2012=> x[1+p/100] And finally at the end of 2013 => x[1+p/100][1-12/100]	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
As per question=> Price was simple 10 percent greater Hence x[1+10/100] must be the final price. Equating the two we get => x[110/100]=x[1+p/100][88/100] => 44p+4400=5500 => 44p=1100 => p=1100/44=> 100/4=> 25. So p must be 25 _________________ Intern Joined: 02 Aug 2016 Posts: 4 Re: The price of a consumer good increased by p%. . . [#permalink] ### Show Tags 16 Nov 2016, 09:46 [size=150]Let Price = 100 [size=150]Increased by P% = 100(1+P/100) Treat it like successive percents; So a 12% decrease would mean 88% of (1+P/100) of 100 The key words are no other change took place. So there are no further successive percents, and the final price = 110 Therefore: 88/100 * (1+P/100) * 100 = 110 => 8800 + 88P = 11,000 => 88P = 2200 => P = 2200/88 => P = 25% e-GMAT Representative Joined: 04 Jan 2015 Posts: 2888 Re: The price of a consumer good increased by p%. . . [#permalink] ### Show Tags Updated on: 18 Dec 2016, 22:12 Hey, The given question can be solved in a number of ways. Let's look at two most common ways to solve this question. We will share two more ways of solving this question tomorrow. Method 1 : • Let us consider the price of the consumer good at the beginning of 2012 to be 100. • Let us also assume the price to be “C” at the beginning of 2013, after an increase of p%. • Since we know that with respect to the initial price, the price at the end of 2013 went up by 10%. o Therefore, the price at the end of 2013 = $$100 + (10$$ % of $$100) = 110$$ • Now we can write - o $$C – 12$$ % of $$C = 110$$ o $$C * (1 - \frac {12}{100}) = 110$$ o $$C = 110 * \frac {25}{22} = 125$$ Therefore, the price at the beginning of 2013 is 125 and we got this value after p% increase over the initial value. Thus, we can write – • $$100 + (p$$ % of $$100) = 125$$ • $$P = 25$$ % Method 2 : Conventional method –	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
Method 2 : Conventional method – • Let the price of consumer good at the beginning of 2012 be 100. • After an increase of p%, the price at the beginning of 2013 will be – o Price at the beginning of 2013 $$= 100 + (p$$ % of $$100) = 100 + p$$ Therefore the price at the beginning of 2013 is (100 + p) • After a decrease of 12%, the price at the end of 2013 will be – o Value at the beginning of 2013 $$* (1 – \frac{12}{100}) = (100 + p) * \frac {22}{25}$$……………..(i) • And we are also given that the overall increase in the price of consumer good is 10%. • Therefore, the value at the end of 2013 = $$100 + (10$$% of $$100)$$ = 110………(ii) • Equating equation (i) and (ii) we get – o $$(100 + p) * \frac {22}{25} = 110$$ o $$100 + p = 125$$ Therefore, $$p = 25$$% There are more innovative ways to solve this question. A few of them have not been discussed here. Can you all think of any other way to solve it? Would love to see a few other methods! I will post two more ways to solve this question tomorrow. In the mean time, expecting some more responses with other ways to solve this question Thanks, Saquib e-GMAT Quant Expert _________________ Originally posted by EgmatQuantExpert on 15 Dec 2016, 23:26. Last edited by EgmatQuantExpert on 18 Dec 2016, 22:12, edited 1 time in total. e-GMAT Representative Joined: 04 Jan 2015 Posts: 2888 Re: The price of a consumer good increased by p%. . . [#permalink] ### Show Tags 16 Dec 2016, 04:11 1 joannaecohen wrote: Lets start with a number for the original value. 100 is the easiest. So we're looking for a value of 110 at the end of 2013. Just by looking at the values we can get an idea of what to start testing. If we're increasing 100 by p% then decreasing it by 12% and the original value is still 10% higher we need a value much higher than 12. 25% is the easiest value to start with. 100+25%=125 125-12%=110 D Hi,	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
100+25%=125 125-12%=110 D Hi, Thanks for posting a different way of approaching this problem. In fact, the approach followed by you is very close to one of the innovative ways that we talked about in our official solution. The only difference being in your approach you have concluded that p should be much larger than 12. Going a step further, you can also conclude that p should be larger than 22% (12%+10%). Once you do so, you don't even need to pick any number. The only option which will satisfy it is D. 25%. When we post our detailed solution using the two innovative methods tomorrow, we will explain how we can conclude that p should be greater than 22%. Regards, Saquib _________________ e-GMAT Representative Joined: 04 Jan 2015 Posts: 2888 Re: The price of a consumer good increased by p%. . . [#permalink] ### Show Tags 25 Dec 2016, 07:32 1 As discussed, let's look at one of the innovative ways of solving the above question. It is one of the quickest ways to solve a question that involves successive percentage increase/decrease on the same value. Please take a note of this approach and apply it on some GMAT questions to master it. So, let's quickly look at this smart approach. When a number is increased successively by two percentage, let's assume, $$a$$% and $$b$$%, the net increase in the value of the number can be expressed by the formula, Net increase $$=a+b+\frac {ab}{100}$$ Le's take a simple example to understand. If we increase a number, let's say, X successively by 10% and 20% respectively, the net increase according to the above formula should be, Net increase $$=10+20+\frac {10*20}{100}=10+20+\frac {200}{100} = 10+20+2 = 32$$% Isn't that quick!! A nice method to keep in your arsenal to solve Percent question involving successive increase quickly.	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
One good thing about the above formula is that you can use it to calculate the net decrease in case of successive decrease too. All you need to do is in case of decrease represent the percent as negative. Easy isn't it . Let's see an application quickly. If we decrease a number, let's say, X successively by 10% and 20% respectively, the net increase according to the above formula should be, Net increase $$=(-10)+(-20)+\frac {(-10)(-20)}{100}=-10-20+\frac {200}{100} = -10-20+2 = (-28)$$% Notice carefully, the sign of the net increase is negative, clearly indicating the after the successive decrease the value of the original number, decreased instead of increasing. And what was the magnitude??? Right 28%. The net decrease is 28%. So, before we use this approach to give you an official answer for the above question, would you like to have a quick stab at it. Remember, you need to be careful about the sign of the change. Increase is represented by positive and decrease is represented by negative. All the best. We will post the detailed solution tomorrow and then we will show another innovative method of solving this question. Regards, Saquib _________________ e-GMAT Representative Joined: 04 Jan 2015 Posts: 2888 The price of a consumer good increased by p%. . . [#permalink] ### Show Tags Updated on: 07 Aug 2018, 07:11 2 1 Alright, so let's look at the official solution to the above questions using the innovative formula on Net increase discussed in the last post. We know that the price of the consumer good increased by $$p$$% and then decreased by $$12$$%. Hence, using the formula for net increase we can say, Net increase $$=p+(-12)+\frac {p(-12)}{100}=p-12-\frac {3p}{25} = (\frac {22p}{25} - 12)$$% It is given in the question that the net increase finally is $$10$$%. Hence, we can equate the two values. $$(\frac {22p}{25} - 12)$$% = $$10$$% or, $$\frac {22p}{25} = 10+12 = 22$$% or, $$p = 25$$%	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
$$(\frac {22p}{25} - 12)$$% = $$10$$% or, $$\frac {22p}{25} = 10+12 = 22$$% or, $$p = 25$$% Now, with this understanding try to solve this question in an even better way. Give it a try and we will post the official solution in another innovative way soon. _________________ Originally posted by EgmatQuantExpert on 27 Dec 2016, 00:45. Last edited by EgmatQuantExpert on 07 Aug 2018, 07:11, edited 1 time in total. Board of Directors Status: QA & VA Forum Moderator Joined: 11 Jun 2011 Posts: 4499 Location: India GPA: 3.5 Re: The price of a consumer good increased by p%. . . [#permalink] ### Show Tags 27 Dec 2016, 07:56 EgmatQuantExpert wrote: The price of a consumer good increased by $$p$$% during $$2012$$ and decreased by $$12$$% during $$2013$$. If no other change took place in the price of the good and the price of the good at the end of $$2013$$ was $$10$$% higher than the price at the beginning of $$2012$$, what was the value of $$p$$? A. $$-2$$% B. $$2$$% C. $$22$$% D. $$25$$% E. Cannot be determined $$p - 12 - \frac{12p}{100} = 10$$ $$100p - 1200 -12p = 1000$$ $$88p = 2200$$ $$p = 25$$ Hence, the correct answer must be (D) 25 _________________ Thanks and Regards Abhishek.... PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS How to use Search Function in GMAT Club \| Rules for Posting in QA forum \| Writing Mathematical Formulas \|Rules for Posting in VA forum \| Request Expert's Reply ( VA Forum Only ) Target Test Prep Representative Status: Founder & CEO Affiliations: Target Test Prep Joined: 14 Oct 2015 Posts: 6522 Location: United States (CA) Re: The price of a consumer good increased by p%. . . [#permalink] ### Show Tags	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
### Show Tags 01 Feb 2019, 18:59 EgmatQuantExpert wrote: The price of a consumer good increased by $$p$$% during $$2012$$ and decreased by $$12$$% during $$2013$$. If no other change took place in the price of the good and the price of the good at the end of $$2013$$ was $$10$$% higher than the price at the beginning of $$2012$$, what was the value of $$p$$? A. $$-2$$% B. $$2$$% C. $$22$$% D. $$25$$% E. Cannot be determined We let the 2012 price = n and thus, the price at the end of 2013 will be: (1 + p/100)(0.88)(n) Since the price at the end of 2013 was 10% higher than at the beginning of 2012, we can create the equation: 1.1n = (1 + p/100)(0.88)(n) 1 = (1 + p/100)(0.8) 1 = 0.8 + 0.8p/100 Multiplying by 100, we have: 100 = 80 + 0.8p 20 = 0.8p 25 = p Alternate Solution: Let’s let the price at the beginning of 2012 be 100. Since the price at the end of 2013 was 10% higher, the price at the end of 2013 is 1.1 x 100 = 110. We know the price decreased by 12% and became 110; therefore, before decreasing by 12%, the price was 110/0.88 = 125. Now, the price of 100 increases by p percent and becomes 125; therefore the value of p is [(125 - 100)/100] x 100 = 25. _________________ # Scott Woodbury-Stewart Founder and CEO Scott@TargetTestPrep.com 122 Reviews 5-star rated online GMAT quant self study course See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews If you find one of my posts helpful, please take a moment to click on the "Kudos" button. VP Joined: 09 Mar 2018 Posts: 1004 Location: India Re: The price of a consumer good increased by p%. . . [#permalink] ### Show Tags	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
### Show Tags 01 Feb 2019, 21:46 EgmatQuantExpert wrote: The price of a consumer good increased by $$p$$% during $$2012$$ and decreased by $$12$$% during $$2013$$. If no other change took place in the price of the good and the price of the good at the end of $$2013$$ was $$10$$% higher than the price at the beginning of $$2012$$, what was the value of $$p$$? A. $$-2$$% B. $$2$$% C. $$22$$% D. $$25$$% E. Cannot be determined 10 % will be a successive percentage change, which is calculated by a + b + ab/100 = Percentage change p - 12 - 12p/100 = 10 100p -1200 - 12p = 1000 p = 2200/88 p = 25 % D _________________ If you notice any discrepancy in my reasoning, please let me know. Lets improve together. Quote which i can relate to. Many of life's failures happen with people who do not realize how close they were to success when they gave up. Re: The price of a consumer good increased by p%. . . [#permalink] 01 Feb 2019, 21:46 Display posts from previous: Sort by	{ "domain": "gmatclub.com", "id": null, "lm_label": "1. Yes\n2. Yes", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9744347905312774, "lm_q1q2_score": 0.8961316836701599, "lm_q2_score": 0.9196425377849806, "openwebmath_perplexity": 1808.9536375803552, "openwebmath_score": 0.7690556645393372, "tags": null, "url": "https://gmatclub.com/forum/the-price-of-a-consumer-good-increased-by-p-228709.html" }
The idea is to minimize the norm of the difference between the given function and the approximation. Picture: geometry of a least-squares solution. As a result we should get a formula y=F(x), named the empirical formula (regression equation, function approximation), which allows us to calculate y for x's not present in the table. Learn to turn a best-fit problem into a least-squares problem. FINDING THE LEAST SQUARES APPROXIMATION We solve the least squares approximation problem on only the interval [−1,1]. We use the Least Squares Method to obtain parameters of F for the best fit. obtained as measurement data. Because the least-squares fitting process minimizes the summed square of the residuals, the coefficients are determined by differentiating S with respect to each parameter, and setting the result equal to zero. The method of least square • Above we saw a discrete data set being approximated by a continuous function • We can also approximate continuous functions by simpler functions, see Figure 3 and Figure 4 Lectures INF2320 – p. 5/80 Given a function and a set of approximating functions (such as the monomials ), for each vector of numbers define a functional By … The least squares method is one of the methods for finding such a function. Active 7 months ago. Learn examples of best-fit problems. Thus, the empirical formula "smoothes" y values. The radial basis function (RBF) is a class of approximation functions commonly used in interpolation and least squares. Approximation of a function consists in finding a function formula that best matches to a set of points e.g. Least Square Approximation for Exponential Functions. Section 6.5 The Method of Least Squares ¶ permalink Objectives. Ask Question Asked 5 years ago. Vocabulary words: least-squares solution. Orthogonal Polynomials and Least Squares Approximations, cont’d Previously, we learned that the problem of nding the polynomial f n(x), of degree n, that best approximates a function f(x) on an interval	{ "domain": "ac.th", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9875683498785867, "lm_q1q2_score": 0.8960328289423238, "lm_q2_score": 0.9073122169746364, "openwebmath_perplexity": 896.7067055485361, "openwebmath_score": 0.710901141166687, "tags": null, "url": "http://centrallab.msu.ac.th/2f1iaj16/lhcbx.php?aea000=least-square-approximation-of-a-function" }
of nding the polynomial f n(x), of degree n, that best approximates a function f(x) on an interval [a;b] in the least squares sense, i.e., that minimizes kf n fk= Z … ... ( \left[ \begin{array}{c} a \\ b \end{array} \right] \right)\$ using the original trial function. Approximation problems on other intervals [a,b] can be accomplished using a lin-ear change of variable. Recipe: find a least-squares solution (two ways). The RBF is especially suitable for scattered data approximation and high dimensional function approximation. Free Linear Approximation calculator - lineary approximate functions at given points step-by-step This website uses cookies to ensure you get the best experience. Quarteroni, Sacco, and Saleri, in Section 10.7, discuss least-squares approximation in function spaces such as . In this section, we answer the following important question: ∂ S ∂ p 1 = − 2 ∑ i = 1 n x i (y i − (p 1 x i + p 2)) = 0 ∂ S ∂ p 2 = − 2 ∑ i … Least squares method, also called least squares approximation, in statistics, a method for estimating the true value of some quantity based on a consideration of errors in observations or measurements. The smoothness and approximation accuracy of the RBF are affected by its shape parameter. The least squares method is the optimization method. '' y values [ −1,1 ] approximate Functions at given points step-by-step website... The best fit the following important question: least Square approximation for Exponential Functions solution two. For the best fit least Square approximation for Exponential Functions two ways ): find a least-squares (. Saleri, in section 10.7, discuss least-squares approximation in function spaces such as between the function. We solve the least squares approximation we solve the least squares method is one of difference... Given points step-by-step this website uses cookies to ensure you get the best experience such a.. And Saleri, in section 10.7, discuss least-squares approximation in function spaces such as	{ "domain": "ac.th", "id": null, "lm_label": "1. YES\n2. YES", "lm_name": "Qwen/Qwen-72B", "lm_q1_score": 0.9875683498785867, "lm_q1q2_score": 0.8960328289423238, "lm_q2_score": 0.9073122169746364, "openwebmath_perplexity": 896.7067055485361, "openwebmath_score": 0.710901141166687, "tags": null, "url": "http://centrallab.msu.ac.th/2f1iaj16/lhcbx.php?aea000=least-square-approximation-of-a-function" }

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