9-5 Content Standard Geometric Series A.SSE.4 Derive the formula for the sum of a geometric series (when the common ratio is not 1), and use the formula to solve problems. Objective To define geometric series and find their sums What number 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 The symbol means there is no upper limit on the values of n. They go on forever. MATHEMATICAL goes into the box to make the sum 1? What sums do the other nine numbers give? Explain your reasoning. You can write any whole number that has the same digit in every place as the sum of the Y PRACTICES terms of a geometric sequence. For example, Le Lesson Vocabulary Vo g geometric series ge converge diverge 4444 5 4(10)0 1 4(10)1 1 4(10)2 1 4(10)3 You can write any rational number as an infinite repeating decimal. For example, 47 90 5 0.5222 c Therefore, you can write any rational number as a number plus the sum of an infinite geometric sequence. 0.5222 c 5 0.5 1 2(0.1)2 1 2(0.1)3 1 2(0.1)4 1 c Essential Understanding Just as with finite arithmetic series, you can find the sum of a finite geometric series using a formula. You need to know the first term, the number of terms, and the common ratio. A geometric series is the sum of the terms of a geometric sequence. Key Concept Sum of a Finite Geometric Series The sum Sn of a finite geometric series a1 1 a1r 1 a1r2 1 c 1 a1r n21, r 2 1, is a1 A1 2 rn B where a1 is the first term, r is the common ratio, and n is the number of terms. Lesson 9-5 Geometric Series 0595_hsm11a2se_cc_0905.indd 595 595 4/28/11 1:31:57 PM
Problem 1 Finding the Sums of Finite Geometric Series What is the sum of the finite geometric series? A 3 1 6 1 12 1 24 1... 1 3072 What do you need to find the sum? You need the first term, the common ratio, and the number of terms in the series. 24 The first term is 3. The common ratio is 63 5 12 6 5 12 5 2. The nth term is 3072. an 5 a1rn21 3072 5 3? 1024 5 Use the explicit formula. 2n21 Substitute 3 for a1, 2 for r, and 3072 for an. 2n21 Divide each side by 3. 1024 is 210, so n 2 1 5 10 and n 5 11. a 1 A 1 2 rn B Use the sum formula. S11 5 3 A 1 2 211 B 122 Substitute 3 for a1, 2 for r, and 11 for n. 5 6141 Simplify. The sum of the series is 6141. 1 n B a 4Q2 R 20 n50 When the lower limit on n is not 1, how can you find the number of terms? The number of terms always equals upper limit 2 lower limit 11. 0 The first term is a1 5 4 Q 12 R 5 4. The common ratio is r 5 12. The lower limit is 0 and the upper limit is 20, so the number of terms is n 5 21. a1 A1 2 rn B Use the sum formula. 21 S21 5 4a1 2 Q 12 R b 1 2 12 <8 Substitute 4 for a1, 12 for r, and 21 for n. Use a calculator. The sum of the series is approximately 8. Got It? 1. What is the sum of the finite geometric series? a. 215 1 30 2 60 1 120 2 240 1 480 10 b. a 5? (22)n21 See Problem 2 for the outcome. 596 Chapter 9 0595_hsm11a2se_cc_0905.indd 596 Sequences and Series 4/28/11 1:32:03 PM
Problem 2 Using the Geometric Series Formula According to the story on the preceding page, how many total kernels of wheat did the soldier request? The amount of wheat in the first 4 squares Step 1 T total amount of The wheat w U the sum formula to find the total Use amount of wheat. am Identify the first term, common ratio, and the number of terms. a1 5 1, r 5 2, n 5 64 Step 2 Use the sum formula. a1(1 2 rn) Write the sum formula. S64 5 1(1 2 264) 122 Substitute for a1, r, and n. 5 264 2 1 < 1.845 3 1019 Simplify. There will be approximately 1.845 3 1019 kernels of wheat. Got It? 2. To save money for a vacation, you set aside $100. For each month thereafter, you plan to set aside 10% more than the previous month. How much money will you save in 12 months? The terms of a geometric series grow rapidly when the common ratio is greater than 1. Likewise, they diminish rapidly when the common ratio is between 0 and 1. In fact, they diminish so rapidly that an infinite geometric series has a finite sum. Key Concept Infinite Geometric Series An infinite geometric series with first term a1 and common ratio u r u, 1 has a finite sum a S 5 1 21 r. An infinite geometric series with u r u $ 1 does not have a finite sum. Lesson 9-5 Geometric Series 0595_hsm11a2se_cc_0905.indd 597 597 4/28/11 4:25:37 PM
To say that an infinite series a 1 1 a 2 1 a 3 1 c,has a sum means that the sequence of partial sums S 1 5 a 1, S 2 5 a 1 1 a 2, S 3 5 a 1 1 a 2 1 a 3, c, S n 5 a 1 1 a 2 1 c 1 a n, cconverges to a number S as n gets very large. When an infinite series does not converge to a sum, the series diverges. An infinite geometric series with u r u $ 1 diverges. When does an infinite geometric series converge? An infinite geometric series converges when the absolute value of the common ratio is less than 1. Problem 3 Analyzing Infinite Geometric Series Does the series converge or diverge? If it converges, what is the sum? A 1 1 1 2 1 1 4 1... r 5 1 2 4 1 5 1 2 1 Since u r u 5 P 2 P, 1, the series converges. S 5 a 1 1 2 r 5 1 1 2 1 5 1 5 2 1 2 2 B àn50 Q 2 3 RQ25 4 Rn Since u r u 5 P2 5 4 P 5 5 4. 1, the series diverges. Got It? 3. Does the infinite series converge or diverge? If it converges, what is the sum? a. 1 2 1 3 4 1 9 8 1 c b. 1 3 2 1 9 1 1 27 2 1 81 1 c c. ` a Q 2 3 Rn d. Reasoning Will an infinite geometric series either converge or diverge? Explain. Lesson Check Do you know HOW? Evaluate each finite geometric series. 1. 1 5 1 1 10 1 1 20 1 1 40 1 1 80 2. 9 2 6 1 4 2 8 3 1 16 9 Determine whether each infinite geometric series diverges or converges. 3. 1 2 1 6 1 1 36 2 1 216 1 c 4. 1 64 1 1 32 1 1 16 1 c Do you UNDERSTAND? MATHEMATICAL PRACTICES 5. Error Analysis A classmate uses the formula for the sum of an infinite geometric series to evaluate 1 1 1.1 1 1.21 1 1.331 1 c and gets 210. What error did your classmate make? 6. Writing Explain how you can determine whether an infinite geometric series has a sum. 7. Compare and Contrast How are the formulas for the sum of a finite arithmetic series and the sum of a finite geometric series similar? How are they different? 598 Chapter 9 Sequences and Series
Practice and Problem-Solving Exercises MATHEMATICAL PRACTICES A Practice Evaluate the sum of the finite geometric series. 8. 1 1 2 1 4 1 8 1 c 1 128 See Problem 1. 9. 4 1 12 1 36 1 108 1 c 1 972 10. 3 1 6 1 12 1 24 1 48 1 c 1 768 11. 25 2 10 2 20 2 40 2 c 2 2560 5 12. a 3 n 4 13. a Q 1 n11 2 R 4 14. a Q 2 n21 5 3 R 15. a Q 1 n21 3 R 16. Financial Planning In March, a family starts saving for a vacation they are planning for the end of August. The family expects the vacation to cost $1375. They start with $125. Each month they plan to deposit 20% more than the previous month. Will they have enough money for their trip? If not, how much more do they need? Determine whether each infinite geometric series diverges or converges. If the series converges, state the sum. See Problem 2. See Problem 3. 17. 1 1 1 4 1 1 16 1 c 18. 1 2 1 2 1 1 4 2 c 19. 4 1 2 1 1 1 c 20. 1 1 2 1 4 1 c 21. 6 1 18 1 54 1 c 22. 254 2 18 2 6 2 c 23. 1 2 1 1 1 2 c 24. 1 1 1 5 1 1 25 1 c 25. 1 4 1 1 2 1 1 1 2 1 c B Apply Evaluate each infinite geometric series. 26. 1.1 1 0.11 1 0.011 1 c 27. 1.1 2 0.11 1 0.011 2 c 28. 1 2 1 5 1 1 25 2 1 125 1 c 29. 3 1 1 1 1 3 1 1 9 1 c 30. 3 1 2 1 4 3 1 8 9 1 c 31. 3 2 2 1 4 3 2 8 9 1 c Determine whether each series is arithmetic or geometric. Then evaluate the finite series for the specified number of terms. 32. 2 1 4 1 8 1 16 1 c; n 5 10 33. 2 1 4 1 6 1 8 1 c; n 5 20 34. 25 1 25 2 125 1 625 2 c; n 5 9 35. 6.4 1 8 1 10 1 12.5 1 c; n 5 7 36. 1 1 2 1 3 1 4 1 c; n 5 1000 37. 81 1 27 1 9 1 3 1 c; n 5 200 38. Think About a Plan The height a ball bounces is less than the height of the previous bounce due to friction. The heights of the bounces form a geometric sequence. Suppose a ball is dropped from one meter and rebounds to 95% of the height of the previous bounce. What is the total distance traveled by the ball when it comes to rest? Does the problem give you enough information to solve the problem? How can you write the general term of the sequence? What formula should you use to calculate the total distance? Lesson 9-5 Geometric Series 599
39. Communications Many companies use a telephone chain to notify employees of a closing due to bad weather. Suppose a company s CEO (Chief Executive Officer) calls four people. Then each of these people calls four others, and so on. a. Make a diagram to show the first three stages in the telephone chain. How many calls are made at each stage? b. Write the series that represents the total number of calls made through the first six stages. c. How many employees have been notified after stage six? 40. Graphing Calculator The graph models the sum of the first n terms in the geometric series with a 1 5 20 and r 5 0.9. a. Write the first four sums of the series. b. Use the graph to evaluate the series to the 47th term. c. Write and evaluate the formula for the sum of the series. d. Graph the formula using the window values shown. Use the graph to verify your answer to part (b). Evaluate each infinite series that has a sum. X=47 Y=198.58607 Xmin=0 Xmax=94 Xscl=10 Ymin=0 Ymax=250 Yscl=50 41. à Q 1 5 Rn21 42. à 3Q 1 4 Rn21 43. à Q2 1 3 Rn21 44. à 7(2) n21 45. à (20.2) n21 46. Open-Ended Write an infinite geometric series that converges to 3. Use the formula to evaluate the series. 47. Reasoning Find the specified value for each infinite geometric series. a. a 1 5 12, S 5 96; find r b. S 5 12, r 5 1 6 ; find a 1 48. Writing Suppose you are to receive an allowance each week for the next 26 weeks. Would you rather receive (a) $1000 per week or (b) $.02 the first week, $.04 the second week, $.08 the third week, and so on for the 26 weeks? Justify your answer. STEM 49. The sum of an infinite geometric series is twice its first term. a. Error Analysis A student says the common ratio of the series is 3 2. What is the student s error? b. Find the common ratio of the series. 50. Physics Because of friction and air resistance, each swing of a pendulum is a little shorter than the previous one. The lengths of the swings form a geometric sequence. Suppose the first swing of a pendulum has a length of 100 cm and the return swing is 99 cm. a. On which swing will the arc first have a length less than 50 cm? b. What is the total distance traveled by the pendulum when it comes to rest? 51. Where did the formula for summing finite geometric series come from? Suppose the geometric series has first term a 1 and constant ratio r, so that S n 5 a 1 1 a 1 r 1 a 1 r 2 1 c 1 a 1 r n21. a. Show that rs n 5 a 1 r 1 a 1 r 2 1 a 1 r 3 1 c 1 a 1 r n. b. Use part (a) to show that S n 2 rs n 5 a 1 2 a 1 r n. c. Use part (b) to show that S n 5 a 1 2 a 1 rn 1 2 r 5 a 1 (1 2 rn ) 1 2 r. 600 Chapter 9 Sequences and Series
C Challenge 52. The function S(n) 5 10A1 2 0.8n B 0.2 represents the sum of the first n terms of an infinite geometric series. a. What is the domain of the function? b. Find S (n) for n 5 1, 2, 3, c, 10. Sketch the graph of the function. c. Find the sum S of the infinite geometric series. 53. Use the formula for the sum of an infinite geometric series to show that 0.9 5 1. (Hint: 0.9 5 9 10 1 9 100 1 9 1000 5 c) SAT/ACT Standardized Test Prep 5 54. What is the value of a (2n 2 3)? 55. Evaluate the infinite geometric series 2 5 1 25 4 1 8 125 1 c. Enter your answer as a fraction. 56. Use log 5 2 < 0.43 and log 5 7 < 1.21 and the properties of logarithms to approximate log 5!14 without using a calculator. 57. Use a calculator to solve the equation 7 2x 5 75. Round the answer to the nearest hundredth. 58. Use the Change of Base Formula and a calculator to solve log 9 x 5 log 6 15. Round the answer to the nearest tenth. Mixed Review Evaluate each series to the given term. 59. 12.5 1 15 1 17.5 1 20 1 22.5 1 c; 7th term 60. 2100 2 95 2 90 2 85 2 c; 11th term Add or subtract. Simplify where possible. 61. 7 2c 2 2 c 2 62. 5 y 1 3 1 15 y 2 3 63. 4 x 2 2 36 1 x x 2 6 See Lesson 9-4. See Lesson 8-5. Use the properties of logarithms to evaluate each expression. 64. log 2 1 8 1 log 2 8 65. log 15 25 1 log 15 9 66. 3 log 9 3 2 1 4 log 9 81 See Lesson 7-4. Get Ready! To prepare for Lesson 10-1, do Exercises 67 69. Graph each function. See Lesson 4-2. 67. y 5 x 2 2 4 68. y 5 x 2 2 6x 2 9 69. y 524x 2 1 1 Lesson 9-5 Geometric Series 601