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the exponent must be n + 3. an = en + 3 13.4 Write an explicit formula for the nth term of the sequence. {9, − 81, 729, − 6,561, 59,049, …} 13.5 Write an explicit formula for the nth term of the sequence. ⎧ ⎨− 3 4 ⎩, − 9 8, − 27 12, − 81 16, − 243 20 ⎫,... ⎬ ⎭ 1454 Chapter 13 Sequences, Probability, and Counting Theor... |
2a3 − 1 = 2(9) − 1 = 17 So the first four terms of the sequence are {3, 5, 9, 17}. The recursive formula for the Fibonacci sequence states the first two terms and defines each successive term as the sum of the preceding two terms. a1 = 1 a2 = 1 an = an − 1 + an − 2, for n ≥ 3 To find the tenth term of the sequence, fo... |
− 20 = − 17 a5 = 3a4 − 20 = 3( − 17) − 20 = − 51 − 20 = − 71 The first five terms are {9, 7, 1, – 17, – 71}. See Figure 13.6. Figure 13.6 13.7 Write the first five terms of the sequence defined by the recursive formula. a1 = 2 an = 2an − 1 + 1, for n ≥ 2 1456 Chapter 13 Sequences, Probability, and Counting Theory Give... |
positive integers from 1 to n. For example = 24 = 120 1457 is the An example of formula containing a factorial is an = (n + 1)!. The sixth term of the sequence can be found by substituting 6 for n. The factorial of any whole number n is n(n − 1)! We can therefore also think of 5! as 5 ⋅ 4!. a6 = (6 + 1 = 5040 Factoria... |
decreasing and nearing zero. Figure 13.8 13.9 Write the first five terms of the sequence defined by the explicit formula an = (n + 1)! 2n. Access this online resource for additional instruction and practice with sequences. • Finding Terms in a Sequence (http://openstaxcollege.org/l/findingterms) This content is availa... |
� 2 ⎩ n if n is not divisible by 4 an = an = ⎧ ⎨ ⎩ −0.6 ⋅ 5 n − 1 2.5 ⋅ ( − 2) if n is prime or 1 n − 1 if n is composite ⎧ ⎨ ⎩ 4(n2 − 2) if n ≤ 3 or n > 6 n2 − 2 4 if 3 < n ≤ 6 For the following exercises, write an explicit formula for each sequence. 21. 22. 23. 24. 25. 4, 7, 12, 19, 28, … −4, 2, − 10, 14, − 34, … 1, ... |
⎝−an − 1 + 1⎞ ⎠ 2 50. an = 1, an = an − 1 + 8 For the following exercises, write a recursive formula for each sequence. 51. an = (n + 1)! (n − 1)! 34. 35. 36. 37. 38. −2.5, − 5, − 10, − 20, − 40, … −8, − 6, − 3, 1, 6, … 2, 4, 12, 48, 240, … 35, 38, 41, 44, 47, … 15, 3, 3 5, 3 25, 3 125, ⋯ For the following exercises, ... |
of terms the sequence. Use the >Frac feature to 59. Find a1 = 2, the first five terms of the sequence an = 2 [(an − 1) − 1] + 1. 60. Find the a1 = 8, an = ⎛ ten first ⎝an − 1 + 1⎞ ⎠! an − 1!. terms of the sequence Find 61. a1 = 2, an = nan − 1 the tenth term of the sequence Follow these steps to evaluate a finite sequ... |
2.4n. of the sequence 58. Find a1 = 625, 15th the an = 0.8an − 1 + 18. term of the sequence an = 64. 1462 List an = five terms of the sequence Chapter 13 Sequences, Probability, and Counting Theory the first 15n ⋅ (−2) n − 1 47 65. List an = 5.7 n the four first + 0.275(n − 1)! terms of the sequence 66. List the first... |
truck for $8,000. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years. The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years. In this section, we will consider... |
is arithmetic, do we have to subtract every term from the following term to find the common difference? No. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common difference. 13.10 Is the given sequence arithmetic? If so, find... |
an = a1 + (n − 1)d. Example 13.10 Writing Terms of Arithmetic Sequences 1466 Chapter 13 Sequences, Probability, and Counting Theory Given a1 = 8 and a4 = 14, find a5. Solution The sequence can be written in terms of the initial term 8 and the common difference d. {8, 8 + d, 8 + 2d, 8 + 3d} We know the fourth term equa... |
ERROR: type should be string, got " https://cnx.org/content/col11758/1.5 Chapter 13 Sequences, Probability, and Counting Theory 1467 Example 13.11 Writing a Recursive Formula for an Arithmetic Sequence Write a recursive formula for the arithmetic sequence. { − 18, − 7, 4, 15, 26, …} Solution The first term is given as −18. The common difference can be found by subtracting the first term from the second term. Substitute the initial term and the common difference into the recursive formula for arithmetic sequences. d = −7 − (−18) = 11 a1 = − 18 an = an − 1 + 11, for n ≥ 2 Analysis We see that the common difference is the slope of the line formed when we graph the terms of the sequence, as shown in Figure 13.11. The growth pattern of the sequence shows the constant difference of 11 units. Figure 13.11 Do we have to subtract the first term from the second term to find the common difference? No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference. 13.14 Write a recursive formula for the arithmetic sequence. {25, 37, 49, 61, …} Using Explicit Formulas for Arithmetic Sequences We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept. To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence. an = a1 + d(n − 1) 1468 Chapter 13 Sequences, Probability, and Counting Theory The common difference is −50, so the sequence represents a linear function with a slope of −50. To find the y intercept, we subtract −50 from 200 : 200 − ( − 50) = 200 + 50 = 250. You can also find the y -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in Figure 13.12. Figure 13.12 Recall the slope-intercept form of a line is y = mx + b. When" |
dealing with sequences, we use an in place of y and n in place of x. If we know the slope and vertical intercept of the function, we can substitute them for m and b in the slopeintercept form of a line. Substituting − 50 for the slope and 250 for the vertical intercept, we get the following equation: an = − 50n + 250 ... |
the common difference and the first term into an = a1 + d(n – 1). 3. Substitute the last term for an and solve for n. Example 13.13 Finding the Number of Terms in a Finite Arithmetic Sequence Find the number of terms in the finite arithmetic sequence. {8, 1, –6,..., –41} 1470 Chapter 13 Sequences, Probability, and Cou... |
71 The child’s allowance at age 16 will be $23 per week. A11 = 1 + 2(11) = 23 13.17 A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from tod... |
is 5, common difference is 6, find the 8th 87. term. First term is 6, common difference is 7, find the 6th 88. term. First term is 7, common difference is 8, find the 7th 89. term. For the following exercises, find the first term given two terms from an arithmetic sequence. Find the first term or a1 90. a6 = 12 and a1... |
. 106. an = ⎧ ⎨ − 0.52, − 1.02, − 1.52,...⎫ ⎬ ⎭ ⎩ an = ⎧ ⎨1 5 ⎩, 9 20, 7 10 ⎫,... ⎬ ⎭ 107. an = ⎧ ⎨−,... ⎬ ⎭ 108. an = ⎧ ⎨1 6 ⎩, − 11 12 ⎫, − 2,... ⎬ ⎭ 124. 125. 126. an = ⎧ ⎨3, − 4, − 11,..., − 60⎫ ⎬ ⎭ ⎩ an = ⎧ ⎨1.2, 1.4, 1.6,..., 3.8⎫ ⎬ ⎭ ⎩ an = ⎧ ⎨1 2 ⎩, 2, 7 2,..., 8 ⎫ ⎬ ⎭ Graphical For the following exercises, det... |
⎨1.8, 3.6, 5.4,...⎫ ⎬ ⎭ ⎩ an = ⎧ ⎨−18.1, −16.2, −14.3,...⎫ ⎬ ⎭ ⎩ an = ⎧ ⎨15.8, 18.5, 21.2,...⎫ ⎬ ⎭ ⎩ an = ⎧ ⎨1 3 ⎩, − 4 3, −3,... ⎫ ⎬ ⎭ 122. an = ⎧ ⎨0, 1 3 ⎩ ⎫,... ⎬ ⎭, 2 3 128. 1474 Chapter 13 Sequences, Probability, and Counting Theory ◦ Set TblStart = 1 ◦ Set ΔTbl = 1 ◦ Set Indpnt: Auto and Depend: Auto • Press [2N... |
the 5th term of 139. {9b, 5b, b, … }. the arithmetic sequence • Press [MODE] ◦ Select SEQ in the fourth line ◦ Select DOT in the fifth line ◦ Press [ENTER] • Press [Y=] ◦ ◦ ◦ nMin is the first counting number for the sequence. Set nMin = 1 u(n) is the pattern for the sequence. Set u(n) = 3n − 2 u(nMin) is the sequence... |
. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In thi... |
of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not. Figure 13.14 If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio? No. If you know that the se... |
1 = − 10 a3 = − 2a2 = 20 a4 = − 2a3 = − 40 The first four terms are {5, –10, 20, –40}. 13.20 List the first five terms of the geometric sequence with a1 = 18 and r = 1 3. Using Recursive Formulas for Geometric Sequences A recursive formula allows us to find any term of a geometric sequence by using the previous term. E... |
� ⎭ Using Explicit Formulas for Geometric Sequences Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms. 1480 Chapter 13 Sequences, Probability, and C... |
} Solution The first term is 2. The common ratio can be found by dividing the second term by the first term. The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula. 10 2 = 5 The graph of this sequence in Figure 13.17 shows an exponential pattern. an = a1 r (n − 1) n − 1 a... |
://openstaxcollege.org/l/geometricseq) • Determine the Type of Sequence (http://openstaxcollege.org/l/sequencetype) • Find the Formula for a Sequence (http://openstaxcollege.org/l/sequenceformula) 1484 Chapter 13 Sequences, Probability, and Counting Theory 13.3 EXERCISES Verbal 146. What is a geometric sequence? How is... |
the first five terms of the geometric sequence, given the first term and common ratio. 159. a1 = 8, r = 0.3 160. a1 = 5, r = 1 5 169. an = {−1, 5, − 25, 125,...} 170. an = {−32, − 16, − 8, − 4,...} 171. an = {14, 56, 224, 896,...} 172. an = {10, − 3, 0.9, − 0.27,...} 173. an = {0.61, 1.83, 5.49, 16.47,...} 174. an = ⎧... |
�−1, − 4 5 ⎩, − 16 25, − 64 125 ⎫,... ⎬ ⎭ 185. an = 186. an = ⎧ ⎨2, 1 3 ⎩, 1 18, 1 108 ⎫,... ⎬ ⎭ 192. ⎧ ⎨3, − 1, 1 3 ⎩, − 1 9 ⎫,... ⎬ ⎭ For the following exercises, find the specified term for the geometric sequence given. 187. Let a1 = 4, an = − 3an − 1. Find a8. 188. Let an = − n − 1 ⎛ ⎝− 1 3 ⎞ ⎠. Find a12. For the f... |
�� ⎠ 2 3 first have a non-integer value? the geometric sequence 203. Use the recursive formula to write a geometric sequence whose common ratio is an integer. Show the first four terms, and then find the 10th term. Use the explicit 204. formula to write a geometric sequence whose common ratio is a decimal number betwee... |
Σ, to represent the sum. Summation notation includes an explicit formula and specifies the first and last terms in the series. An explicit formula for each term of the series is given to the right of the sigma. A variable called the index of summation is written below the sigma. The index of summation is set equal to ... |
= 7. We find the terms of the series by substituting k = 3,4,5,6, and 7 into the function k 2. We add the terms to find the sum. 7 ∑ k = 3 k 2 = 32 + 42 + 52 + 62 + 72 = 9 + 16 + 25 + 36 + 49 = 135 13.25 5 Evaluate ∑ k = 2 (3k – 1). Using the Formula for Arithmetic Series Just as we studied special types of sequences,... |
Given terms of an arithmetic series, find the sum of the first n terms. 1. Identify a1 and an. 2. Determine n. 3. Substitute values for a1, an, and n into the formula Sn = n(a1 + an) 2. 4. Simplify to find Sn. Example 13.22 Finding the First n Terms of an Arithmetic Series Find the sum of each arithmetic series. a. b.... |
69 13.28 10 ∑ k = 1 5 − 6k This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 13 Sequences, Probability, and Counting Theory 1491 Example 13.23 Solving Application Problems with Arithmetic Series On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she ... |
(1 − r)Sn = a1 − r n a1 Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for Sn, divide both sides by (1 − r). Sn = a1( 1492 Chapter 13 Sequences, Probability, and Counting Theory Formula for the Sum of the First n Terms o... |
partial sum of each geometric series. 13.30 S20 for the series 1,000 + 500 + 250 + … 13.31 8 ∑ k = 1 k 3 Example 13.25 Solving an Application Problem with a Geometric Series At a new job, an employee’s starting salary is $26,750. He receives a 1.6% annual raise. Find his total earnings at the end of 5 years. Solution ... |
than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with −1 < r < 1 approach 0; the sum of a geometric series is defined when −1 < r < 1. 1494 Chapter 13 Sequences, Probability, and Counting Theory Determining Whe... |
13.35 ∞ ∑ k = 1 15 ⋅ ( – 0.3) k Finding Sums of Infinite Series When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first n terms of a geometric series. We will examine an infinite series with r = 1 2.... |
3 c. The formula is exponential, so the series is geometric with r = – 1 3. Find a1 by substituting k = 1 into the given explicit formula: a1 = 4,374 ⋅ ( – 1 3 ) 1 – 1 = 4,374 Substitute a1 = 4,374 and r = − 1 3 into the formula, and simplify to find the sum: S = a1 1 − r S = 4,374 1 − ( − 1 3) = 3,280.5 d. The formul... |
initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the acc... |
the month. How much is in the account right after the last deposit? Solution The value of the initial deposit is $100, so a1 = 100. A total of 120 monthly deposits are made in the 10 years, so n = 120. To find r, divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new mon... |
= − 2 to k = 1 The sum that results from adding the number 4 five 214. times For the following exercises, express each arithmetic sum using summation notation. 224. 225. 226 11 ∑ a = 1 n − 1 5 ⋅ 2 64 ⋅ 0.2 a − 1 For the following exercises, determine whether the infinite series has a sum. If so, write the formula for ... |
.2 + 3.5 + 4.8 6 + 15 2 + 9 + 21 2 + 12 + 27 2 + 15 −1 + 3 + 7 +... + 31 11 ∑ For the following exercises, use the formula for the sum of the first n terms of a geometric series to find the partial sum. 243. 244. 245. 246. S6 S7 for the series −2 − 10 − 50 − 250... for the series 0.4 − 2 + 10 − 50... 9 ∑ k = 1 10 ∑ 2 ⋅... |
261. To get the best loan rates available, the Riches want to save enough money to place 20% down on a $160,000 home. They plan to make monthly deposits of $125 in an investment account interest compounded semi-annually. Will the Riches have enough for a 20% down payment after five years of saving? How much money will... |
utations involving n non-distinct objects. A new company sells customizable cases for tablets and smartphones. Each case comes in a variety of colors and can be personalized for an additional fee with images or a monogram. A customer can choose not to personalize or could choose to have one, two, or three images or a m... |
possible choices of one each as shown in the tree diagram in Figure 13.19. Figure 13.19 The possible choices are: 1. 2. 3. 4. 5. 6. 7. 8. 9. soup, chicken, cake soup, chicken, pudding soup, fish, cake soup, fish, pudding soup, steak, cake soup, steak, pudding salad, chicken, cake salad, chicken, pudding salad, fish, c... |
we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall. There are four options for the first place, so we write a 4 on the first line. This content i... |
6 for the fourth, and so on until only 1 person remains for the last spot. There are 362,880 possible permutations for the swimmers to line up. Analysis Note that in part c, we found there were 9! ways for 9 people to line up. The number of permutations of n distinct objects can always be found by n!. A family of five... |
that the formula stills works if we are choosing all n objects and placing them in order. In that case we would be dividing by (n − n)! or 0!, which we said earlier is equal to 1. So the number of permutations of n objects taken n at a time is n! 1 or just n!. Formula for Permutations of n Distinct Objects Given n dis... |
we are selecting objects and the order does not matter, we are dealing with combinations. A selection of r objects from a set of n objects where the order does not matter can be written as C(n, r). Just as with permutations, C(n, r) can also be written as n Cr. In this case, the general formula is as follows. C(n, r) ... |
3) = 5! 3!(5 − 3)! = 10 Analysis We can also use a graphing calculator to find combinations. Enter 5, then press n Cr, enter 3, and then press the equal sign. The n Cr, function may be located under the MATH menu with probability commands. Is it a coincidence that parts (a) and (b) in Example 13.34 have the same answe... |
, and sour cream as toppings for a baked potato. How many different ways are there to order a potato? Solution We are looking for the number of subsets of a set with 4 objects. Substitute n = 4 into the formula. There are 16 possible ways to order a potato. n 2 = 24 = 16 A sundae bar at a wedding has 6 toppings to choo... |
13.49 Find the number of rearrangements of the letters in the word CARRIER. Access these online resources for additional instruction and practice with combinations and permutations. • Combinations (http://openstaxcollege.org/l/combinations) • Permutations (http://openstaxcollege.org/l/permutations) 1512 Chapter 13 Seq... |
an odd number from A? set How How many ways are there to pick a red ace or a club 275. from a standard card playing deck? How many ways are there to pick a paint color from 5 276. shades of green, 4 shades of blue, or 7 shades of yellow? How many outcomes are possible from tossing a pair 277. of coins? How many outcom... |
same set. What is the value of n? (Hint: the order in which the elements for the subsets are chosen is not important.) 304. Can C(n, r) ever equal P(n, r)? Explain. Suppose a set A has 2,048 subsets. How many 305. distinct objects are contained in A? Hector wants to place billboard advertisements throughout the county... |
6 roses, and 8 daisies? How many unique ways can a string of Christmas 321. lights be arranged from 9 red, 10 green, 6 white, and 12 gold color bulbs? Real-World Applications A family consisting of 2 parents and 3 children is to 307. pose for a picture with 2 family members in the front and 3 in the back. a. How many ... |
, r) = n! r!(n − r)! n The combination ⎛ r ⎝ ⎞ ⎠ is called a binomial coefficient. An example of a binomial coefficient is ⎛ 5 ⎝ 2 ⎞ ⎠ = C(5, 2) = 10. Binomial Coefficients If n and r are integers greater than or equal to 0 with n ≥ r, then the binomial coefficient is n ⎛ r ⎝ ⎞ ⎠ = C(n, r) = n! r!(n − r)! Is a binomial... |
for finding more complicated binomial expansions. (x + y)2 = x2 + 2xy + y2 (x + y)3 = x3 + 3x2 y + 3xy2 + y3 (x + y)4 = x4 + 4x3 y + 6x2 y2 + 4xy3 + y4 First, let’s examine the exponents. With each successive term, the exponent for x decreases and the exponent for y increases. The sum of the two exponents is n for eac... |
each term is n. • The powers on x begin with n and decrease to 0. • The powers on y begin with 0 and increase to n. • The coefficients are symmetric. To determine the expansion on (x + y)5, we see n = 5, degree of 5. In descending order for powers of x, the pattern is as follows: thus, there will be 5+1 = 6 terms. Eac... |
�� ⎞ ⎠xyn − 1 + yn ⎝ n − 1 (13.12) Given a binomial, write it in expanded form. 1. Determine the value of n according to the exponent. 2. Evaluate the k = 0 through k = n using the Binomial Theorem formula. 3. Simplify. Example 13.38 Expanding a Binomial Write in expanded form. a. (x + y)5 1518 Chapter 13 Sequences, Pr... |
This will occur whenever the binomial contains a subtraction sign. 13.51 Write in expanded form. a. b. (x − y)5 (2x + 5y)3 Using the Binomial Theorem to Find a Single Term Expanding a binomial with a high exponent such as (x + 2y)16 can be a lengthy process. Sometimes we are interested only in a certain term of a bino... |
)9 = 5,857,280x7 y9 13.52 Find the sixth term of (3x − y)9 without fully expanding the binomial. Access these online resources for additional instruction and practice with binomial expansion. • The Binomial Theorem (http://openstaxcollege.org/l/binomialtheorem) • Binomial Theorem Example (http://openstaxcollege.org/l/b... |
, find the indicated term of each binomial without fully expanding the binomial. 351. 352. 353. 354. 355. The fourth term of (2x − 3y)4 The fourth term of (3x − 2y)5 The third term of (6x − 3y)7 The eighth term of (7 + 5y)14 The seventh term of (a + b)11 356. The fifth term of (x − y)7 The tenth term of (x − 1)12 The n... |
of an − k bk is the same as the coefficient of which other term? n 368. Consider the expansion of (x + b)40. What is the exponent of b in the kth term? 369. n ⎛ ⎝ k n ⎞ ⎞ Find ⎛ ⎠ and write ⎠ + ⎝ k − 1 n ⎞ binomial coefficient in the form ⎛ ⎠. Prove it. Hint: Use ⎝ k integer p, such that answer fact any the as a that,... |
certain event. A probability model is a mathematical 1. The figure is for illustrative purposes only and does not model any particular storm. This content is available for free at https://cnx.org/content/col11758/1.5 Chapter 13 Sequences, Probability, and Counting Theory 1523 description of an experiment listing all p... |
event by the total number of outcomes in S. Suppose a number cube is rolled, and we are interested in finding the probability of the event “rolling a number less than or equal to 4.” There are 4 possible outcomes in the event and 6 possible outcomes in S, so the probability of the event is 4 6 = 2 3. Computing the Pro... |
is both orange and a b twice. To find the probability of spinning an orange or a b, we need to subtract the probability that the sector is both orange and has a b The probability of spinning orange or a b is 2 3. Probability of the Union of Two Events The probability of the union of two events E and F (written E ∪ F )... |
with mutually exclusive events, the intersection of E and F is the empty set. The probability of spinning an orange is 3 6 and the probability of spinning a d is 1 6. We can find the probability of spinning an orange or a d simply = 1 2 by adding the two probabilities. P(E ∪ F) = P(E) + P(F The probability of spinning... |
a probability model must be 1. P(E′) = 1 − P(E) The probability of the horse winning added to the probability of the horse losing must be equal to 1. Therefore, if the probability of the horse winning the race is 1 9, the probability of the horse losing the race is simply 1 − 1 9 = 8 9 The Complement Rule The probabil... |
Rule to find the probability that the sum is less than 13.57 10. Computing Probability Using Counting Theory Many interesting probability problems involve counting principles, permutations, and combinations. In these problems, we will use permutations and combinations to find the number of elements in events and sampl... |
to choose 3 dogs. Since we are choosing both bears and dogs at the same time, we will use the Multiplication Principle. There are C(6, 2) ⋅ C(5, 3) ways to choose 2 bears and 3 dogs. We can use this result to find the probability. C(6,2)C(5,3) C(14,5) = 15 ⋅ 10 2,002 = 75 1,001 c. It is often easiest to solve “at leas... |
online resources for additional instruction and practice with probability. • Introduction to Probability (http://openstaxcollege.org/l/introprob) • Determining Probability (http://openstaxcollege.org/l/determineprob) this website (http://openstaxcollege.org/l/PreCalcLPC11) Visit Learningpod. for additional practice qu... |
the following exercises, two coins are tossed. 384. What is the sample space? 396. A club 397. A two 398. Six or seven 399. Red six 400. An ace or a diamond 401. A non-ace 402. A heart or a non-jack For the following exercises, two dice are rolled, and the results are summed. Construct a table showing the sample space... |
: 413. A head on the coin or a club 414. A tail on the coin or red ace 415. A head on the coin or a face card 416. No aces For the following exercises, use this scenario: a bag of M&Ms contains 12 blue, 6 brown, 10 orange, 8 yellow, 8 red, and 4 green M&Ms. Reaching into the bag, a person grabs 5 M&Ms. 417. What is the... |
does not matter common difference the difference between any two consecutive terms in an arithmetic sequence common ratio the ratio between any two consecutive terms in a geometric sequence complement of an event the set of outcomes in the sample space that are not in the event E diverge a series is said to diverge if... |
specifies the first and last terms in the series term a number in a sequence union of two events the event that occurs if either or both events occur upper limit of summation the number used in the explicit formula to find the last term in a series KEY EQUATIONS Formula for a factorial (n − 1)(n − 2) ⋯ (2)(1), for n ≥... |
formulas define each term of a sequence using the position of the term. See Example 13.1, Example 13.2, and Example 13.3. • An explicit formula for the nth term of a sequence can be written by analyzing the pattern of several terms. See Example 13.4. • Recursive formulas define each term of a sequence using previous t... |
with common ratio r is given by an = ran – 1 for n ≥ 2. • As with any recursive formula, the initial term of the sequence must be given. See Example 13.17. • An explicit formula for a geometric sequence with common ratio r is given by an = a1 r n – 1. See Example 13.19. In application problems, we sometimes alter the ... |
solved using the Multiplication Principle or the formula for P(n, r). See Example 13.32 and Example 13.33. • A selection of objects where the order does not matter is a combination. • Given n distinct objects, the number of ways to select r objects from the set is C(n, r) and can be found using a formula. See Example ... |
the sequence defined by n the explicit formula an = 10 + 3. 1538 Chapter 13 Sequences, Probability, and Counting Theory 434. Write the first four terms of the sequence defined by n! the explicit formula an = n(n + 1). Arithmetic Sequences 435. Is the sequence 4 7, 47 21 find the common difference., 82 21, 39 7,... ari... |
, 6, 3, 3 2, … Geometric Sequences 443. Find the common ratio for the geometric sequence 2.5, 5, 10, 20, … 444. Is the sequence 4, 16, 28, 40, … geometric? If so find the common ratio. If not, explain why. 445. A geometric sequence has terms a7 = 16,384 and a9 = 262,144. What are the first five terms? 446. A geometric ... |
,32} that is divisible by either 4 or 6? In a group of 20 musicians, 12 play piano, 7 play 462. trumpet, and 2 play both piano and trumpet. How many musicians play either piano or trumpet? 470. A day spa charges a basic day rate that includes use of a sauna, pool, and showers. For an extra charge, guests can choose fro... |
? How 469. {1, 3, 5, …, 99} have? many subsets does the set 480. What is the probability that a roll includes a 2 or results in a pair? 481. What is the probability that a roll doesn’t include a 2 or result in a pair? 482. What is the probability of rolling a 5 or a 6? 483. What is the probability that a roll includes ... |
. 494. Is the sequence − 2, − 1, − 1 2, − 1 4, … geometric? If so find the common ratio. If not, explain why. 495. What is the 11th term of the geometric sequence − 1.5, − 3, − 6, − 12, …? 496. Write a recursive formula for the geometric sequence 1, … 497. Write an explicit formula for the geometric sequence 4, − 4 3, ... |
has 8 candy toppings and 4 fruit toppings to choose from. How many ways are there to top a frozen yogurt? How many 508. the word EVANESCENCE be arranged if the anagram must end with the letter E? distinct ways can 509. Use the Binomial Theorem to expand ⎛ ⎝ 3 2 x − 1 2 5. y⎞ ⎠ 510. Find the seventh term of ⎛ ⎝x2 − 1 2... |
logb a Heron’s Formula A = s(s − a)(s − b)(s − c) 1544 where s = a + b + c 2 Proof: Let a, b, and c be the sides of a triangle, and h be the height. Appendix A Figure A1 So s = a + b + c 2. We can further name the parts of the base in each triangle established by the height such that p + q = c. Figure A2 Using the Pyt... |
4c2 (a + b + c)(a + c − b)(b + a − c)(b − a + c) 4c2 (a + b + c)( − a + b + c)(a − b + c)(a + b − c) 4c2 2s ⋅ (2s − a) ⋅ (2s − b)(2s − c) 4c2 = = = = = Therefore, h2 = h = 4s(s − a)(s − b)(s − c) c2 2 s(s − a)(s − b)(s − c) c And since A = 1 2 ch, then A = 1 2 c2 s(s − a)(s − b)(s − c) c = s(s − a)(s − b)(s − c) Prope... |
u1 w1, u2 w2,..., un wn 〉 = 〈 u1, u2,...un 〉 · 〈 v1, v2,...vn 〉 + 〈 u1, u2,...un 〉 · 〈 w1, w2,...wn 〉 = u|2 Proof: 1546 Appendix A u · u = 〈 u1, u2,...un 〉 · 〈 u1, u2,...un 〉 = u1 u1 + u2 u2 +... + un un 2 + u2 2 +... + un = u1 = | 〈 u1, u2,...un 〉 |2 = v · u 2 Standard Form of the Ellipse centered at the Origin y2 b2... |
2 + y2 = 4a2 − 4a (x − c)2 + y2 + x2 − 2cx + y2 2cx = 4a2 − 4a (x − c)2 + y2 − 2cx − 1 4a ⎠ = (x − c)2 + y2 4cx − 4a2 = 4a (x − c)2 + y2 ⎛ ⎝4cx − 4a2⎞ c ax = (x − c)2 + y2 x2 = (x − c)2 + y2 a2 − 2xc + a2 − 2xc + a − c2 a2 c2 a2 c2 a2 c2 a2 a2 + a2 + x2 = x2 − 2xc + c2 + y2 x2 = x2 + c2 + y2 x2 = x2 + c2 + y2 Let 1 = a... |
�� ⎝y − 0⎞ ⎠ 2 = 2a This content is available for free at https://cnx.org/content/col11758/1.5 Appendix A 1549 (x − ( − c))2 + (y − 0)2 − (x − c)2 + (y − 0)2 = 2a (x + c)2 + y2 − (x − c)2 + y2 = 2a (x + c)2 + y2 = 2a + (x − c)2 + y2 ⎝2a + (x − c)2 + y2⎞ ⎛ (x + c)2 + y2 = ⎠ x2 + 2cx + c2 + y2 = 4a2 + 4a (x − c)2 + y2 x2... |
) a2 b2 = b2 x2 − a2 y2 a2 y2 a2 b2 a2 b2 = a2 b2 b2 x2 a2 b2 − y2 x2 a2 − b2 Trigonometric Identities 1 = Pythagorean Identity Even-Odd Identities Table A1 cos2 t + sin2 t = 1 1 + tan2 t = sec2 t 1 + cot2 t = csc2 t cos( − t) = cos t sec( − t) = sec t sin( − t) = − sin t tan( − t) = − tan t csc( − t) = − csc t cot( − ... |
± 1 − cos α 1 + cos α = sin α 1 + cos α = 1 − cos α sin α Cofunction Identities Fundamental Identities Sum and Difference Identities Double-Angle Formulas Half-Angle Formulas Table A1 This content is available for free at https://cnx.org/content/col11758/1.5 Appendix A 1551 Reduction Formulas sin2 θ = cos2 θ = tan2 θ ... |
2 = a2 + c2 − 2ac cos β c2 = a2 + b2 − 2ab cos γ 1552 Appendix A ToolKit Functions Figure A7 Figure A8 This content is available for free at https://cnx.org/content/col11758/1.5 Appendix A 1553 Figure A9 Trigonometric Functions Unit Circle Figure A10 1554 Appendix A Angle 0 π 6, or 30 ° π 4, or 45 ° π 3, or 60 ° π 2, o... |
average rate of change, 282, 379 axes of symmetry, 1365 axis of symmetry, 476, 481, 634, 1395, 1397 B base, 16, 99 binomial, 68, 99, 552 binomial coefficient, 1514, 1533 binomial expansion, 1515, 1518, 1533 Binomial Theorem, 1517, 1533 break-even point, 1225, 1331 C cardioid, 1118, 1201 carrying capacity, 762, 793 Car... |
66 cosecant, 874, 892, 932 cosecant function, 932, 933, 979 cosine, 1022, 1024 cosine function, 851, 892, 903, 905, 907, 917 cost function, 300, 1225, 1331 cotangent, 873, 892, 940 cotangent function, 940, 979 coterminal angles, 821, 824, 892 Cramer’s Rule, 1318, 1321, 1326, 1331 cube root, 500 cubic functions, 612 cur... |
ellipsis, 1446 end behavior, 502, 592, 634 endpoint, 285, 810 entry, 1273, 1331 equation, 26, 99, 235 equation in quadratic form, 199 equation in two variables, 111, 216 equations in quadratic form, 216 Euler, 1134 even function, 334, 379, 978 even-odd identities, 977, 1049 event, 1522, 1533 experiment, 1522, 1533 exp... |
, 793 Heaviside method, 1262 Heron of Alexandria, 1086 Heron’s formula, 1086 horizontal asymptote, 583, 590, 634 horizontal compression, 340, 379, 1042 horizontal line, 140, 418, 466 horizontal line test, 244, 379 horizontal reflection, 328, 379 horizontal shift, 321, 379, 671, 702, 904 horizontal stretch, 340, 379 hyp... |
, 1317 irrational numbers, 12, 15, 100 J joint variation, 629, 634 K Kronecker, 1134 L latus rectum, 1387, 1395, 1437 Law of Cosines, 1080, 1201 Law of Sines, 1062, 1081, 1201 leading coefficient, 68, 100, 507, 634 leading term, 68, 100, 507, 635 least common denominator, 92, 100, 131 least squares regression, 455, 466... |
, 1406, 1437 nondegenerate conic sections, 1404 nonlinear inequality, 1252, 1331 nth term of the sequence, 1446 nth partial sum, 1533 nth root of a complex number, 1144 nth term of a sequence, 1533 nth partial sum, 1487 O oblique triangle, 1060, 1201 odd function, 334, 379, 977 one-loop limaçon, 1120 one-loop limaҫon, ... |
omial equation, 192, 217 polynomial function, 506, 523, 533, 539, 635 position vector, 1179, 1181 positive angle, 812, 892 power function, 500, 635 power rule for logarithms, 726, 732, 793 principal nth root, 61, 100 principal square root, 54, 100 probability, 1522, 1534 probability model, 1522, 1534 product of two mat... |
rectangular form, 1137, 1168 recursive formula, 1454, 1466, 1478, 1534 reduction formulas, 1012, 1049 reference angle, 822, 863, 877, 892 reflection, 676, 711 regression analysis, 774, 778, 781 regression line, 456 relation, 226, 380 remainder, 551 Remainder Theorem, 562, 635 removable discontinuity, 588, 635 Restrict... |
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