884 CHAPTER 12 Sequences; Induction; the Binomial Theorem In Problems 19–26, find the fifth term and the nth term of the geometric sequence whose first term a1 and common ratio r are given. 19. a r 2; 3 1 = = 20. a r 2; 4 1 = − = 21. a r 5; 1 1 = = − 22. a r 6; 2 1 = = − 23. a r 0; 1 7 1 = = 24. = = − a r 1; 1 3 1 25. a r 3; 3 1 = = 26. a r 0; 1 1 π = = In Problems 27–32, find the indicated term of each geometric sequence. 27. 7th term of … 1, 1 2 , 1 4 , 28. 8th term of 1, 3, 9, ... 29. 15th term of … 1, 1, 1, − 30. 10th term of … 1, 2, 4, − − 31. 8th term of 0.4, 0.04, 0.004, ... 32. 7th term of 0.1, 1.0, 10.0, ... In Problems 33–40, find the nth term an of each geometric sequence. When given, r is the common ratio. 33. 6, 18, 54, 162, ... 34. 5, 10, 20, 40, ... 35. 3, 1, 1 3 , 1 9 , − − … 36. … 4, 1, 1 4 , 1 16 , 37. a r 243; 3 6 = = − 38. a r 7; 1 3 2 = = 39. a a 7; 1575 2 4 = = 40. a a 1 3 ; 1 81 3 6 = = In Problems 41–46, find each sum. 41. 1 4 2 4 2 4 2 4 2 4 n 2 3 1 + + + + + − 42. 3 9 3 9 3 9 3 9 n 2 3 + + + + 43. 2 3 k n k 1 ∑( ) = 44. 4 3 k n k 1 1 ∑ ⋅ = − 45. 1 2 4 8 2n 1 ( ) − − − − − − − 46. 2 6 5 18 25 2 3 5 n 1 ( ) + + + + − For Problems 47–52, use a graphing utility to find the sum of each geometric sequence. 47. 1 4 2 4 2 4 2 4 2 4 2 3 14 + + + + + 48. 3 9 3 9 3 9 3 9 2 3 15 + + + + 49. 2 3 n n 1 15 ∑( ) = 50. 4 3 n n 1 15 1 ∑ ⋅ = − 51. 1 2 4 8 214 − − − − − − 52. 2 6 5 18 25 2 3 5 15 ( ) + + + + In Problems 53–68, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. 53. 1 1 3 1 9 + + + 54. 2 4 3 8 9 + + + 55. 8 4 2 + + + 56. 6 2 2 3 + + + 57. 2 1 2 1 8 1 32 − + − + 58. 1 3 4 9 16 27 64 − + − + 59. 8 12 18 27 + + + + 60. 9 12 16 64 3 + + + + 61. 5 1 4 k k 1 1 ∑ ( ) = ∞ − 62. 8 1 3 k k 1 1 ∑ ( ) = ∞ − 63. 1 2 3 k k 1 1 ∑ ⋅ = ∞ − 64. 3 3 2 k k 1 1 ∑ ( ) = ∞ − 65. 6 2 3 k k 1 1 ∑ ( ) − = ∞ − 66. 4 1 2 k k 1 1 ∑ ( ) − = ∞ − 67. 3 2 3 k k 1 ∑ ( ) = ∞ 68. 2 3 4 k k 1 ∑ ( ) = ∞ 83. Find x so that x x , 2, + and x 3 + are consecutive terms of a geometric sequence. 84. Find x so that x 1, − x, and x 2 + are consecutive terms of a geometric sequence. 85. Salary Increases If you have been hired at an annual salary of $42,000 and expect to receive annual increases of 3%, what will your salary be when you begin your fifth year? Applications and Extensions 86. Equipment Depreciation A new piece of equipment cost a company $15,000. Each year, for tax purposes, the company depreciates the value by 15%. What value should the company give the equipment after 5 years? 87. Pendulum Swings Initially, a pendulum swings through an arc of length 2 feet. On each successive swing, the length of the arc is 0.9 of the previous length. Mixed Practice In Problems 69–82, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. 69. n 2 { } + 70. n2 5 { } − 71. { } n4 2 72. n5 1 2 { } + 73. n 3 2 3 { } − 74. n 8 3 4 { } − 75. 1, 3, 6, 10, ... 76. 2, 4, 6, 8, ... 77. 2 3 n { } ( ) 78. 5 4 n { } ( ) 79. − − … 1, 2, 4, 8, 80. 1, 1, 2, 3, 5, 8, ... 81. { } 3n/2 82. 1 n { } ( ) −
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