A-56 Answers to Selected Exercises (b) 0 0 14 21 The population density converges to 9.5 thousand per acre. 97. (a) Nj+1 = 2Nj for j Ú 1 (b) 1840 (c) 0 0 15,000 10 (d) As j becomes large, the values of Nj increase without bound. 99. (a) 0.0198026273; ln 1.02 ≈0.0198026273 (b) -0.0304592075; ln 0.97 ≈ -0.0304592075 101. (a) 2.718254; e ≈2.718282 (b) 0.367857; e-1 ≈0.367879 11.2 Exercises 1. difference 3. 6 5. 51, 2, 3, 4, 56; 5-4, -1, 2, 5, 86 7. 3 9. 55 11. 3 13. -5 15. x + 2y 17. 8, 14, 20, 26, 32 19. 5, 3, 1, -1, -3 21. 10 + 27, 10, 10 - 27, 10 - 227, 10 - 327 23. an = 2n + 3; a8 = 19 25. an = 10 3 n + 5 3 ; a8 = 85 3 27. an = 9 2 n - 39; a8 = -3 29. an = x + 3n - 3; a8 = x + 21 31. an = s + 2pn - 2p; a8 = s + 14p 33. 3 35. 34 37. 2.5 In Exercises 39–43, D is the domain and R is the range. 39. an = n - 3; D: 51, 2, 3, 4, 5, 66; R: 5-2, -1, 0, 1, 2, 36 41. an = - 1 2 n + 3; D: 51, 2, 3, 4, 5, 66; R: 50, 0.5, 1, 1.5, 2, 2.56 43. an = -20n + 30; D: 51, 2, 3, 4, 56; R: 5-70, -50, -30, -10, 106 45. 215 47. 230 49. 160 51. 77.5 53. 55p 55. 3240 57. 2500 59. 3660 61. a1 = 7, d = 5 63. a1 = 5, d = -2 65. a1 = 1, d = - 20 11 67. 18 69. 140 71. -621 73. 500,500 75. 10,100 77. 6150 79. 328.3 81. 172.884 83. 1281 85. 4680 87. 54,800 89. 713 in. 91. yes 11.3 Exercises 1. ratio 3. 24 5. 155 7. geometric; r = -2 9. neither 11. (a) $5.12 (b) $10.23 13. (a) $163.84 (b) $327.67 There are other acceptable forms of the answers in Exercises 85 and 87. 85. a 9 i=1 1 3i 87. a 8 k=1 A - 1 2B k-1 89. converges to 1 2 91. diverges 93. converges to e ≈2.71828 95. (a) a1 = 8 thousand per acre, a2 = 10.4 thousand per acre, a3 = 8.528 thousand per acre In Exercises 15–25, there may be other ways to express an. 15. a5 = 80; an = 51-22n-1 17. a 5 = -108; an = - 4 3 132n-1 19. a 5 = -729; an = -91-32n-1 21. a5 = -324; an = -4132n-1 23. a 5 = 125 4 ; an = 4 5 A 5 2B n-1 25. a5 = 5 8 ; an = 10 A - 1 2B n-1 27. 2; -3 29. 1 5 ; 125 31. {0.1; 5000 33. 682 35. 99 8 37. 860.95 39. -183 41. 189 4 43. 2032 45. The sum exists if r 61. 47. 2; diverges 49. 1 2 53. 27 55. 3 20 57. 4 59. 1 2 61. - 8 15 63. 3 4 65. 97.739 67. 0.212 69. (a) a1 = 1169; r = 0.916 (b) a10 = 531; a20 = 221; A person who is 10 yr from retirement should have savings of 531% of his or her annual salary. A person 20 yr from retirement should have savings of 221% of his or her annual salary. 71. (a) an = a1 # 2n-1 (b) 15 (rounded from 14.29) (c) 560 min, or 9 hr, 20 min 73. 488 75. 62; 2046 77. 1 64 m 79. Option 2 81. $10,159.11 83. $25,423.18 85. $28,107.41 87. $64,060.60 91. yes Summary Exercises on Sequences and Series 1. geometric; r = 2 2. arithmetic; d = 3 3. arithmetic; d = - 5 2 4. neither 5. geometric; r = 4 3 6. geometric; r = -3 7. neither 8. arithmetic; d = -3 9. neither 10. geometric; r = -25 11. geometric; 3122n-1; 3069 12. arithmetic; 4n - 2; 200 13. arithmetic; - 3 2 n + 11 2 ; - 55 2 14. geometric; 3 2 A 2 3B n-1 , or A2 3B n-2 ; 58,025 13,122 15. geometric; 31-22n-1; -1023 16. arithmetic; -3n - 2; -185 17. diverges 18. 1111 500 19. -1850 20. 1092 21. - 4 3 22. diverges 23. 144 24. 1 4 25. diverges 26. 0.3 + 0.03 + 0.003 + g ; 1 3 11.4 Exercises 1. sum 3. 1 5. 8 7. x8; y8 9. 5p4q 11. 20 13. 35 15. 56 17. 45 19. 1 21. n 23. 56 25. 4950 27. 1 29. 12 31. x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + y6 33. p5 - 5p4q + 10p3q2 - 10p2q3 + 5pq4 - q5 35. r10 + 5r8s + 10r6s2 + 10r4s3 + 5r2s4 + s5 37. p4 + 8p3q + 24p2q2 + 32pq3 + 16q4 39. 2401p4 - 2744p3q + 1176p2q2 - 224pq3 + 16q4 41. 729x6 - 2916x5y + 4860x4y2 - 4320x3y3 + 2160x2y4 - 576xy5 + 64y6 43. m6 64 - 3m5 16 + 15m4 16 - 5m3 2 + 15m2 4 - 3m+ 1 45. 4r4 + 8 22r3 m + 12r2 m2 + 422r m3 + 1 m4 47. 1 x16 + 4 x8 + 6 + 4x 8 + x16 49. -3584h3j5
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