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ic in Polar Form to Rectangular Form Convert the conic r = to rectangular form. 1 _________ 5 − 5sin θ Solution We will rearrange the formula to use the identities r = √ — x 2 + y2, x = r cos θ, and y = r sin θ. r = 1 _________ 5 − 5sin θ 1 _________ 5 − 5sin θ r ⋅ (5 − 5 sin θ) = ⋅ (5 − 5 sin θ) 5r − 5r sin θ = 1 5r =...
IC For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. 6. r = 10. r = 6 __________ 1 − 2 cos θ 16 __________ 4 + 3 cos θ 7. r = 11. r = 3 __________ 4 − 4 sin θ 3 ____________ 10 + 10 cos θ 8. r = 12. r = 8 __________ 4 − 3 cos θ 2 ________ 1 − cos θ...
cos θ 33. r = 37. r = 10 __________ 5 − 4 sin θ 2 _ 1 − sin θ 34. r = 38. r = 3 __________ 1 + 2 cos θ 6 __________ 3 + 2 sin θ 39. r(1 + cos θ) = 5 40. r(3 − 4sin θ) = 9 41. r(3 − 2sin θ) = 6 42. r(6 − 4cos θ) = 5 For the following exercises, find the polar equation of the conic with focus at the origin and the given...
of a hyperbola the midpoint of both the transverse and conjugate axes of a hyperbola center of an ellipse the midpoint of both the major and minor axes conic section any shape resulting from the intersection of a right circular cone with a plane conjugate axis the axis of a hyperbola that is perpendicular to the trans...
rix polar equation an equation of a curve in polar coordinates r and θ transverse axis the axis of a hyperbola that includes the foci and has the vertices as its endpoints Key equations Horizontal ellipse, center at origin Vertical ellipse, center at origin Horizontal ellipse, center (h, k) Vertical ellipse, center (h,...
y) in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci). • When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See Example 1 and Example 2. • When given an equatio...
axis as its axis of symmetry can be used to graph the parabola. If p > 0, the parabola opens right. If p < 0, the parabola opens left. See Example 1. • The standard form of a parabola with vertex (0, 0) and the y-axis as its axis of symmetry can be used to graph the parabola. If p > 0, the parabola opens up. If p < 0, ...
it enables us to identify the conic section. See Example 5. 10.5 Conic Sections in Polar Coordinates • Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus P(r, θ) at the pole, and a line, the directrix, which ...
A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be? THe HYPeRBOlA For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, an...
Focus at  2, _ _ 8 8 32. A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep. ROTATIOn OF AxeS For the following exercises, determine which of the conic sections is represen...
a conic with focus at the origin, find the equation in polar form. 49. Directrix is x = 3 and eccentricity e = 1 50. Directrix is y = −2 and eccentricity e = 4 936 CHAPTER 10 analytic geometry CHAPTeR 10 PRACTICe TeST For the following exercises, write the equation in standard form and state the center, vertices, and ...
y + 3) 16. Write the equation of a parabola with a focus at 15. y 2 + 8x − 8y + 40 = 0 17. A searchlight is shaped like a paraboloid of (2, 3) and directrix y = −1. revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the w...
slim, but we all love to fantasize about what we could buy with the winnings. One of the first things a lottery winner has to decide is whether to take the winnings in the form of a lump sum or as a series of regular payments, called an annuity, over the next 30 years or so. This decision is often based on many factor...
the terms for a sequence can be cumbersome. For example, finding the number of hits on the website at the end of the month would require listing out as many as 31 terms. A more efficient way to determine a specific term is by writing a formula to define the sequence. One type of formula is an explicit formula, which d...
term of the sequence, an 1 2 2 4 Table 2 3 8 4 16 5 32 n 2n Graphing provides a visual representation of the sequence as a set of distinct points. We can see from the graph in Figure 1 that the number of hits is rising at an exponential rate. This particular sequence forms an exponential function. an 36 32 28 24 20 16...
have identified all n terms. Example 1 Writing the Terms of a Sequence Defined by an Explicit Formula Write the first five terms of the sequence defined by the explicit formula an = −3n + 8. Solution Substitute n = 1 into the formula. Repeat with values 2 through 5 for n. n = 1 a1 = −3(1) + 8 = 5 n = 2 a2 = −3(2) + 8 ...
each value of n into the formula. Begin with n = 1 to find the first term, a1. The sign of the term is given by the (−1)n in the explicit formula. 2. To find the second term, a2, use n = 2. 3. Continue in the same manner until you have identified all n terms. Example 2 Writing the Terms of an Alternating Sequence Defi...
subsection. How To… Given an explicit formula for a piecewise function, write the first n terms of a sequence 1. Identify the formula to which n = 1 applies. 2. To find the first term, a1, use n = 1 in the appropriate formula. 3. Identify the formula to which n = 2 applies. 4. To find the second term, a2, use n = 2 in...
in the terms. Keep in mind that the pattern may involve alternating terms, formulas for numerators, formulas for denominators, exponents, or bases. How To… Given the first few terms of a sequence, find an explicit formula for the sequence. 1. Look for a pattern among the terms. 2. If the terms are fractions, look for ...
…} 944 CHAPTER 11 seQuences, proBaBility and counting theory Try It #5 Write an explicit formula for the nth term of the sequence. 9 3  − __ __, −, − 8 4 27 ___ 12, −, − 81 ___ 16 243 ___ 20,...  Try It #6 Write an explicit formula for the nth term of the sequence. 1  __ e2, 1 _ e, 1, e, e 2,...  Writing the Terms...
a1 = 1 a2 = 1 an = an − 1 + an − 2 for n ≥ 3 To find the tenth term of the sequence, for example, we would need to add the eighth and ninth terms. We were told previously that the eighth and ninth terms are 21 and 34, so a10 = a9 + a8 = 34 + 21 = 55 recursive formula A recursive formula is a formula that defines each ...
6 (4, −17) n (5, −71) Figure 5 Try It #7 Write the first five terms of the sequence defined by the recursive formula. a1 = 2 an = 2an − 1 + 1, for n ≥ 2 How To… Given a recursive formula with two initial terms, write the first n terms of a sequence. 1. Identify the initial term, a1, which is given as part of the formu...
1 = 24 5 = 120 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)! = 7,040 SECTION 11.1 seQuences and their notations 947 n factori...
, as the plot of the terms shows, the terms are decreasing and nearing zero. 5 . _____ 1,008 1 ___, 36 1, 5 6 an, 5 12 3, 1 8 4, 1 36 5, 5 1008 0 1 2 4 3 Figure 7 n 5 6 Try It #9 Write the first five terms of the sequence defined by the explicit formula an = (n + 1)! _______. 2n Access this online resource for additio...
− 2) if n ≤ 3 or n > 6 2 _ n if n is not divisible by 4 n2 − 2 _ 4 if 3 < n ≤ 6 if n ≤ 5 n2 _ 2n + 1 n2 − 5 if n > 5 17. an = { 19. an = { −0.6 ⋅ 5n − 1 if n is prime or 1 2.5 ⋅ (−2)n − 1 if n is composite For the following exercises, write an explicit formula for each sequence. 21. 4, 7, 12, 19, 28, … 22. −4, 2, − 10...
47, … 3 __ 38. 15, 3,, 5 3 ___, 25 3 ___ 125, … For the following exercises, evaluate the factorial. 39. 6! 40.  12 ! ___ 6 41. 12! ___ 6! 42. 100! ____ 99! For the following exercises, write the first four terms of the sequence. 45. an = 44. an = 43. an = 3 ⋅ n! _____ 4 ⋅ n! n! __ n2 n! _________ n2 − n − 1 46. an ...
defined recursively using a graphing calculator: • On the home screen, key in the value for the initial term a1 and press [ENTER]. • Enter the recursive formula by keying in all numerical values given in the formula, along with the key strokes [2ND] ANS for the previous term an − 1. Press [ENTER]. • Continue pressing ...
. Press [ENTER]. • Enter the items in the order “Expr”, “Variable”, “start”, “end” separated by commas. See the instructions above for the description of each item. • Press [ENTER] to see the list of terms for the finite sequence defined. Use the right arrow key to scroll through the list of terms. For the following ex...
often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same...
the subsequent term to determine whether a common difference exists. a. The sequence is not arithmetic because there is no common difference 16 − 8 = 8 b. The sequence is arithmetic because there is a common difference. The common difference is 4. 1 − (−3 13 − 9 = 4 952 CHAPTER 11 seQuences, proBaBility and counting t...
to find the next term. The first five terms are {17, 14, 11, 8, 5} SECTION 11.2 arithmetic seQuences 953 Analysis As expected, the graph of the sequence consists of points on a line as shown in Figure 2. an 20 16 12 Figure 2 Try It #3 List the first five terms of the arithmetic sequence with a1 = 1 and d = 5. How To… ...
of the previous term and the 954 CHAPTER 11 seQuences, proBaBility and counting theory common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given. an = an − 1 + d n ≥ 2 recursive formula for an arithmetic sequen...
can subtract the common difference from the first term of the sequence. Consider the following sequence. −50 −50 −50 −50 {200, 150, 100, 50, 0,...} 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 ...
vertical intercept of −8. an 50 40 30 20 10 0 –10 1 2 3 4 5 6 7 8 9 10 n Figure 5 Try It #6 Write an explicit formula for the following arithmetic sequence. {50, 47, 44, 41, … } Finding the Number of Terms in a Finite Arithmetic Sequence Explicit formulas can be used to determine the number of terms in a finite arithm...
can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2. Let A be the amount of the allowance and n be the number of years after age 5. Using the altered explicit formula for an arithmetic sequence we get: b. We can find the number of years since age 5 by subtracting. 16 − 5 = 1...
series given two terms. 12. a1 = 17, a7 = −31 13. a13 = −60, a33 = −160 For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. 14. First term is 3, common difference is 4, find the 15. First term is 4, common difference is 5, find the 5th term. 4th ...
, 5 7 9 ___ ___, 20 10 33. a = {8.9, 10.3, 11.7,... } 5 1, −2,...  36. a =  − __ __, − 4 2 11 ___ 12, −2,...  1 37. a =  __, − 6 For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. 38. a = {7, 4, 1,... }; Find the 17th term. 40. a = {2, 6, 10,... }; Find ...
5 1 0.5 0 −0.5 −0.5 −1 −1.5 −2 −2.5 −3 −3.5 −4 −4.5 −5 −5.5 (5, 4) (4, 2) (3, 0) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n (2, −2) (1, −4) 57. an 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 (5, 7.5938) (4, 5.0625) (3, 3.375) (2, 2.25) (1, 1.5) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 n −0.5 0 −0.5 960 CHAPTER 11 seQuences, ...
nMin = 1, nMax = 5, xMin = 0, xMax = 6, yMin = −1, and yMax = 14. Then press [GRAPH]. Graph the sequence as it appears on the graphing calculator. 1 __ For the following exercises, follow the steps given above to work with the arithmetic sequence an = n + 5 using a 2 graphing calculator. 64. What are the first seven t...
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 this section, we will rev...
geometric and so this interpretation holds, but (b) is not. an 20 16 12 8 4 0 an 60 48 36 24 12 1 2 3 4 5 6 (a) n 0 Figure 1 1 2 3 4 5 6 n (b) Q & A… 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 sequence is geomet...
so on. a1 = 5 a2 = −2a1 = −10 a3 = −2a2 = 20 a4 = −2a3 = −40 The first four terms are {5, −10, 20, −40}. Try It #3 1 __ List the first five terms of the geometric sequence with a1 = 18 and r =. 3 Using Recursive Formulas for Geometric Sequences A recursive formula allows us to find any term of a geometric sequence by ...
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. an = a1 r n − 1 Let’s take a look at the sequence {18,...
of the sequence into the formula. The graph of this sequence in Figure 4 shows an exponential pattern. an = a1r (n − 1) an = 2 ⋅ 5n − 1 an 300 250 200 150 100 50 0 1 2 3 4 5 n Figure 4 Try It #6 Write an explicit formula for the following geometric sequence. {–1, 3, –9, 27,...} 966 CHAPTER 11 seQuences, proBaBility an...
67 11.3 SeCTIOn exeRCISeS VeRBAl 1. What is a geometric sequence? 2. How is the common ratio of a geometric sequence found? 3. What is the procedure for determining whether a 4. What is the difference between an arithmetic sequence is geometric? sequence and a geometric sequence? 5. Describe how exponential functions a...
write the first five terms of the geometric sequence. 1 ___ 22. a1 = −486, an = − an − 1 3 23. a1 = 7, an = 0.2an − 1 For the following exercises, write a recursive formula for each geometric sequence. 24. an = {−1, 5, −25, 125,...} 26. an = {14, 56, 224, 896,...} 25. an = {−32, −16, −8, −4,...} 27. an = {10, −3, 0.9,...
, an = −3an − 1. Find a8. 1  43. Let an = −  − __ 3. Find a12. For the following exercises, find the number of terms in the given finite geometric sequence. 44. an = {−1, 3, −9,..., 2187} 1 45. an =  2, 1, __,..., 2 1  ____ 1024 GRAPHICAl For the following exercises, determine whether the graph shown represents a g...
which term does the geometric sequence 58. Use the recursive formula to write a geometric 2  an = −36  __ 3 n − 1 first have a non-integer value? sequence whose common ratio is an integer. Show the first four terms, and then find the 10th term. 59. Use the explicit formula to write a geometric 60. Is it possible for...
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 the lower limit of summation, which is the number used to generate the first term in the series. The number above the sigma, ca...
to find the sum of k 2 from k = 3 to k = 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 2 = 32 + 42 + 52 + 62 + 72 k = 3 = 9 + 16 + 25 + 36 + 49 = 135 Try It #1 Evaluate ∑ k = 2 5 (3k − 1). Using the Formula for Arithmetic Series ...
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 = 4. Simplify to find Sn. n(a1 + an) _. 2 Example 2 Finding the First n Terms of an Arithmetic Series Find the sum of each arithmetic series. a. 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 ...
It #3 Use the formula to find the sum of the arithmetic series. 13 + 21 + 29 + … + 69 Try It #4 Use the formula to find the sum of the arithmetic series. 10 ∑ k = 1 5 − 6k Example 3 Solving Application Problems with Arithmetic Series On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday...
−(ra1 + r 2a1 + r 3a1 +... + r na1) (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(1 − rn) _________ 1 − r r ≠ 1 formula for the sum of the first n te...
#6 Use the formula to find the indicated partial sum of each geometric series. S20 for the series 1,000 + 500 + 250 + … Try It #7 Use the formula to find the indicated partial sum of each geometric series. 8 ∑ 3k k = 1 Example 5 Solving an Application Problem with a Geometric Series At a new job, an employee’s startin...
975 The common ratio r = 0.2. As n gets very large, the values of rn get very small and approach 0. Each successive term affects the sum less 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 app...
... + + + 8 4 2 3 Try It #10 Determine whether the sum of the infinite series is defined. 24 + (−12) + 6 + (−3) +... Try It #11 Determine whether the sum of the infinite series is defined. ∞ ∑ k = 1 15 ⋅ (−0.3)k Finding Sums of Infinite Series When the sum of an infinite geometric series exists, we can calculate the su...
series is not geometric. b. There is a constant ratio; the series is geometric. a1 = 248.6 and r = 99.44 _____ 248.6 = 0.4, so the sum exists. Substitute a1 = 248.6 and r = 0.4 into the formula and simplify to find the sum: S = a1 _ 1 − r S = _ = 414. 3 248.6 _ 1 − 0.4 1 __. Find a1 by substituting k = 1 into the give...
Annuity Problems 2 2 __ __ +... +.76k + 1 ∞ k ∑ 3   − __ 8 k = 1 At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To f...
Determine r. a. Divide the annual interest rate by the number of times per year that interest is compounded. b. Add 1 to this amount to find r. 4. Substitute values for a1, r, and n into the formula for the sum of the first n terms of a geometric series, Sn = a1(1 – rn) _. 1 – r 5. Simplify to find Sn, the value of th...
eS VeRBAl 1. What is an nth partial sum? 3. What is a geometric series? 5. What is an annuity? AlGeBRAIC For the following exercises, express each description of a sum using summation notation. 6. The sum of terms m2 + 3m from m = 1 to m = 5 7. The sum from of n = 0 to n = 4 of 5n 8. The sum of 6k − 5 from k = −2 to k ...
a savings account. He opened the account with an initial deposit of $50. Each month thereafter he increased the previous deposit amount by $20. 26. Graph the arithmetic sequence showing one year of 27. Graph the arithmetic series showing the monthly Javier’s deposits. sums of one year of Javier’s deposits. For the fol...
50; total deposits: 60; interest rate: 47. Deposit amount: $150; total deposits: 24; interest 5%, compounded monthly rate: 3%, compounded monthly 48. Deposit amount: $450; total deposits: 60; interest 49. Deposit amount: $100; total deposits: 120; interest rate: 4.5%, compounded quarterly rate: 10%, compounded semi-ann...
plan to prepare for finals. On the first day, she plans to study for 1 hour, and each successive day she will increase her study time by 30 minutes. How many hours will Keisha have studied after one week? 60. A scientist places 50 cells in a petri dish. Every hour, the population increases by 1.5%. What will the cell ...
ition Principle According to the Addition Principle, if one event can occur in m ways and a second event with no common outcomes can occur in n ways, then the first or second event can occur in m + n ways. Example 1 Using the Addition Principle There are 2 vegetarian entrée options and 5 meat entrée options on a dinner...
second event can occur in n ways after the first event has occurred, then the two events can occur in m × n ways. This is also known as the Fundamental Counting Principle. Example 2 Using the Multiplication Principle Diane packed 2 skirts, 4 blouses, and a sweater for her business trip. She will need to choose a skirt...
2. Determine how many options are left for the second situation. 3. Continue until all of the spots are filled. 4. Multiply the numbers together. SECTION 11.5 counting principles 985 Example 3 Finding the Number of Permutations Using the Multiplication Principle At a swimming competition, nine swimmers compete in a ra...
It #5 A family of five is having portraits taken. Use the Multiplication Principle to find how many ways the family can line up for the portrait if the parents are required to stand on each end. 986 CHAPTER 11 seQuences, proBaBility and counting theory Finding the Number of Permutations of n Distinct Objects Using a F...
n! _______ (n − r)! How To… Given a word problem, evaluate the possible permutations. 1. Identify n from the given information. 2. Identify r from the given information. 3. Replace n and r in the formula with the given values. 4. Evaluate. Example 4 Finding the Number of Permutations Using the Formula A professor is c...
n! _________ r!(n − r)! An earlier problem considered choosing 3 of 4 possible paintings to hang on a wall. We found that there were 24 ways to select 3 of the 4 paintings in order. But what if we did not care about the order? We would expect a smaller number because selecting paintings 1, 2, 3 would be the same as se...
r objects from n objects, we are not choosing (n − r) objects. Therefore, C(n, r) = C(n, n − r). Try It #8 An ice cream shop offers 10 flavors of ice cream. How many ways are there to choose 3 flavors for a banana split? Finding the number of Subsets of a Set We have looked only at combination problems in which we cho...
mutations of n non-Distinct Objects We have studied permutations where all of the objects involved were distinct. What happens if some of the objects are indistinguishable? For example, suppose there is a sheet of 12 stickers. If all of the stickers were distinct, there would be 12! ways to order the stickers. However,...
A and B are non-overlapping. 1. Use the Addition Principle of counting to explain 2. Use the Multiplication Principle of counting to how many ways event A or B can occur. explain how many ways event A and B can occur. Answer the following questions. 3. When given two separate events, how do we know whether to apply th...
3, 3) 21. C(12, 4) 22. C(26, 3) 18. P(9, 6) 23. C(7, 6) 19. P(11, 5) 24. C(10, 3) For the following exercises, find the number of subsets in each given set. 25. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 27. A set containing 5 distinct numbers, 4 distinct 26. {a, b, c, …, z} 28. The set of even numbers from 2 to 28 letters, and 3...
in the front and 3 in the back. a. How many arrangements are possible with no restrictions? b. How many arrangements are possible if the parents must sit in the front? c. How many arrangements are possible if the parents must be next to each other? packages and 8 different data packages. Of those, 3 packages include b...
A motorcycle shop has 10 choppers, 6 bobbers, and 5 café racers—different types of vintage motorcycles. How many ways can the shop choose 3 choppers, 5 bobbers, and 2 café racers for a weekend showcase? 51. Just-For-Kicks Sneaker Company offers an online customizing service. How many ways are there to design a custom ...
a whole number, a binomial coefficient will always be a whole number. Example 1 Finding Binomial Coefficients Find each binomial coefficient. 9  b.  __ 2 5  a.  __ 3 Solution 9  c.  __ 7 Use the formula to calculate each binomial coefficient. You can also use the nCr function on your calculator. n  _ r  = C(n,...
term, the exponent for x decreases and the exponent for y increases. The sum of the two exponents is n for each term. Next, let’s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern . ,...,  ,  ,   2 1 0 These patterns le...
degree (or sum of the exponents) for 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, thus, there will be 5 + 1 = 6 terms. Each term has a combined degree of 5. In ...
the Binomial Theorem The Binomial Theorem is a formula that can be used to expand any binomial. n (x + y)n = ∑ n  x n − k yk  − 2y 2 +... +    xy n − 1 + y n SECTION 11.6 Binomial theorem 995 How To… Given a binomial, write it in expanded form. 1. Determine the value of n according to the exponent. 2. Evaluate th...
It #2 Write in expanded form. a. (x − y)5 b. (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 binomial expansion. We do not need to fully expand a binomial to find a...
://openstaxcollege.org/l/btexample) SECTION 11.6 section exercises 997 11.6 SeCTIOn exeRCISeS VeRBAl 1. What is a binomial coefficient, and how it is 2. What role do binomial coefficients play in a calculated? binomial expansion? Are they restricted to any type of number? 3. What is the Binomial Theorem and what is its...
The fourth term of (3x − 2y)5 33. The eighth term of (7 + 5y)14 35. The fifth term of (x − y)7 37. The ninth term of (a − 3b 2)11 y 9 x  39. The eighth term of  2 _ _ + 2 For the following exercises, use the Binomial Theorem to expand the binomial f (x) = (x + 3)4. Then find and graph each indicated sum on one set o...
− 2x + 1) b. ( √ c. (x 3 + 2y 2 − z)5 2y 3 )12 d. (3x 2 − √ a − 5)8 a + 4 √ — — SECTION 11.7 proBaBility 999 leARnInG OBjeCTIVeS In this section, you will: • Construct probability models. • Compute probabilities of equally likely outcomes. • Compute probabilities of the union of two events. • Use the complement rule t...
where each event is equally likely, construct a probability model. 1. Identify every outcome. 2. Determine the total number of possible outcomes. 3. Compare each outcome to the total number of possible outcomes. Example 1 Constructing a Probability Model Construct a probability model for rolling a single, fair die, wi...
. Example 2 Computing the Probability of an Event with Equally Likely Outcomes A six-sided number cube is rolled. Find the probability of rolling an odd number. Solution The event “rolling an odd number” contains three outcomes. There are 6 equally likely outcomes in the sample space. Divide to find the probability of ...
Events A card is drawn from a standard deck. Find the probability of drawing a heart or a 7. Solution A standard deck contains an equal number of hearts, diamonds, clubs, and spades. So the probability 1 __ of drawing a heart is. There are four 7s in a standard deck, and there are a total of 52 cards. So the probabili...
Given a set of events, compute the probability of the union of mutually exclusive events. 1. Determine the total number of outcomes for the first event. 2. Find the probability of the first event. 3. Determine the total number of outcomes for the second event. 4. Find the probability of the second event. 5. Add the pr...
3. Solution The first step is to identify the sample space, which consists of all the possible outcomes. There are two number cubes, and each number cube has six possible outcomes. Using the Multiplication Principle, we find that there are 6 × 6, or 36 total possible outcomes. So, for example, 1-1 represents a 1 rolle...
There are 5 phones that are not defective, so there are C(5, 2) ways to select 2 phones that are not defective. There are 8 phones, so there are C(8, 2) ways to select 2 phones. The probability of selecting 2 phones that are not defective is: ways to select 2 phones that are not defective ____ = ways to select 2 phone...
, 5) ways to choose toys from the 9 toys that are not dogs. Since there are 14 toys, there are C(14, 5) ways to choose the 5 toys from all of the toys. C(9, 5) _______ C(14,5) = 63 _____ 1,001 If there is 1 dog chosen, then 4 toys must come from the 9 toys that are not dogs, and 1 must come from the 5 dogs. Since we ar...
. How is this similar to the definition used for the union of two events from a probability model? How is it different? 4. What is the difference between events and outcomes? Give an example of both using the sample space of tossing a coin 50 times. nUMeRIC For the following exercises, use the spinner shown in Figure 3...
of rolling a sum of 5 or 6. 42. Find the probability of rolling any sum other than 5 or 6. For the following exercises, a coin is tossed, and a card is pulled from a standard deck. Find the probability of the following: 43. A head on the coin or a club 44. A tail on the coin or red ace 45. A head on the coin or a face...
S. citizens, what is the percent chance that three are elderly? (Round to the nearest tenth of a percent.) chance that four are elderly? (Round to the nearest thousandth of a percent.) 60. It is predicted that by 2030, one in five U.S. citizens will be elderly. How much greater will the chances of meeting an elderly pe...
terms of a series and written below the sigma with the lower limit of summation infinite sequence a function whose domain is the set of positive integers infinite series the sum of the terms in an infinite sequence lower limit of summation the number used in the explicit formula to find the first term in a series Mult...
utations of n non-distinct objects Binomial Theorem (r + 1)th term of a binomial expansion probability of an event with equally likely outcomes Sn = Sn = Sn = + an) n(a1 _ 2 a1(1 − rn) _________ 1 − r a1 1 P(n, r) = n! _ (n − r)! C(n, r) = n! _ r!(n − r)! n! _ r1!r2! … rk! n (x + y)n = ∑ n _  x n − ky k  k k – 0 n _ ...
common difference repeatedly. See Example 2 and Example 3. • A recursive formula for an arithmetic sequence with common difference d is given by an = an − 1 + d, n ≥ 2. See Example 4. • As with any recursive formula, the initial term of the sequence must be given. • An explicit formula for an arithmetic sequence with ...
Example 7, and Example 8. • An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series. See Example 9. 11.5 Counting Principles • If one event can occur in m ways and a second event with no common outcomes can occur in n ...
probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See Example 4. • The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See Example 5. • In some probability problems, we need to use permutations and...
and a9 = 262,144. What are the first five terms? 16. A geometric sequence has the first term a1 = −3 and 1 _ common ratio r =. What is the 8th term? 2 17. What are the first five terms of the geometric sequence a1 = 3, an = 4 ⋅ an − 1? 18. Write a recursive formula for the geometric 1 _ 27 1 1 _ _,, sequence 1, 9 3, …...
. Write a series representing the total distance traveled by the ball, assuming it was initially dropped from a height of 5 feet. What is the total distance? (Hint: the total distance the ball travels on each bounce is the sum of the heights of the rise and the fall.) 30. The twins Sarah and Scott both opened retiremen...

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