1068 CHAPTER 11 Further Topics in Algebra Determine a5 and an for each geometric sequence. See Example 2. 15. a1 = 5, r = -2 16. a1 = 8, r = -5 17. a2 = -4, r = 3 18. a3 = -2, r = 4 19. a4 = 243, r = -3 20. a4 = 18, r = 2 21. -4, -12, -36, -108, c 22. -2, 6, -18, 54, c 23. 4 5 , 2, 5, 25 2 , c 24. 1 2 , 2 3 , 8 9 , 32 27 , c 25. 10, -5, 5 2 , - 5 4 , c 26. 3, - 9 4 , 27 16 , - 81 64 , c Determine r and a1 for each geometric sequence. See Example 3. 27. a2 = -6, a7 = -192 28. a2 = -8, a7 = 256 29. a3 = 5, a8 = 1 625 30. a4 = - 1 4 , a9 = - 1 128 31. a3 = 50, a7 = 0.005 32. a3 = 300, a9 = 100 243 Use the formula for Sn to find the sum of the first five terms of each geometric sequence. In Exercises 37 and 38, round to the nearest hundredth. See Example 5. 33. 2, 8, 32, 128, c 34. 4, 16, 64, 256, c 35. 18, -9, 9 2 , - 9 4 , c 36. 12, -4, 4 3 , - 4 9 , c 37. a1 = 8.423, r = 2.859 38. a1 = -3.772, r = -1.553 Evaluate each sum. See Example 6. 39. a 5 i=1 1-32i 40. a 4 i=1 1-22i 41. a 6 j=1 48a 1 2b j 42. a 5 j=1 243a 2 3b j 43. a10 k=4 2k 44. a 9 k=3 3k 45. Concept Check Under what conditions does the sum of an infinite geometric series exist? 46. The number 0.999c can be written as the sum of the terms of an infinite geometric sequence: 0.9 + 0.09 + 0.009 + g. Here we have a1 = 0.9 and r = 0.1. Use the formula for S∞ to find this sum. Does intuition indicate that this answer is correct? Find r for each infinite geometric sequence. Identify any whose sum diverges. 47. 12, 24, 48, 96, c 48. 2, -10, 50, -250, c 49. -48, -24, -12, -6, c 50. 625, 125, 25, 5, c Work each problem. See Examples 7 and 8. 51. Use lim nS∞ Sn to show that 2 + 1 + 1 2 + 1 4 + gconverges to 4. 52. We determined that 1 + 1 3 + 1 9 + 1 27 + gconverges to 3 2 using an argument involving limits. Use the formula for the sum of the terms of an infinite geometric sequence to obtain the same result.
RkJQdWJsaXNoZXIy NjM5ODQ=