A-57 Answers to Selected Exercises 51. 74,613a12b16 53. 38,760x6y42 55. 90,720x28y12 57. 11 59. exact: 3,628,800; approximate: 3,598,695.619 60. 0.830% 61. exact: 479,001,600; approximate: 475,687,486.5; 0.692% 62. exact: 6,227,020,800; approximate: 6,187,239,475; 0.639%; As n gets larger, the percent error decreases. 11.5 Exercises 1. 3 + 6 + 9 + 12 = 314214 + 12 2 ; true 3. 1 2 + 1 22 + 1 23 + 1 24 = 24 - 1 24 ; true 5. 24 62142; false 7. S1: 1 = 12; S 2: 1 + 3 = 22; S 3: 1 + 3 + 5 = 32; S4: 1 + 3 + 5 + 7 = 42; S 5: 1 + 3 + 5 + 7 + 9 = 52 Although we do not usually give proofs, the answers for Exercises 9 and 17 are given here. 9. (a) 3112 = 3 and 311211 + 12 2 = 6 2 = 3, so Sn is true for n = 1. (b) 3 + 6 + 9 + g+ 3k = 31k21k + 12 2 (c) 3 + 6 + 9 + g+ 31k + 12 = 31k + 1231k + 12 + 14 2 (d) Add 31k + 12 to each side of the equation in part (b). Simplify the expression on the right side to match the right side of the equation in part (c). (e) Because Sn is true for n = 1, and Sn is true for n = k + 1 when it is true for n = k, Sn is true for every positive integer n. 17. (a) 1 1 # 2 = 1 2 and 1 1 + 1 = 1 2 , so Sn is true for n = 1. (b) 1 1 # 2 + 1 2 # 3 + 1 3 # 4 + g+ 1 k1k + 12 = k k + 1 (c) 1 1 # 2 + 1 2 # 3 + g+ 1 1k + 1231k + 12 + 14 = k + 1 1k + 12 + 1 (d) Add the last term on the left of the equation in part (c) to each side of the equation in part (b). Simplify the right side until it matches the right side in part (c). (e) Because Sn is true for n = 1, and Sn is true for n = k + 1 when it is true for n = k, Sn is true for every positive integer n. For Exercises 21–31, we show only the proof for Exercise 25. 25. (a) 1am21 = am and am112 = am, so S n is true for n = 1. (b) 1am2k = amk (c) 1am21k+12 = am1k+12 (d) 1am2k # 1am21 = amk # 1am21 1am21k+12 = a1mk+m2 Product rule for exponents 1am21k+12 = am1k+12 Factor. (e) Because Sn is true for n = 1, and Sn is true for n = k + 1 when it is true for n = k, Sn is true for every positive integer n. 35. 4n-1 3n-2, or 3 A 4 3B n-1 37. 2n - 1 Chapter 11 Quiz [11.1–11.3] 1. -2, -6, -10, -14, -18; arithmetic 2. 1, - 1 2 , 1 4 , - 1 8 , 1 16 ; geometric 3. 5, 3, 18, 27, 81; neither [11.2] 4. 12 [11.2, 11.3] 5. (a) 430 (b) - 1705 128 6. (a) -1215 (b) diverges (c) 3 [11.4] 7. x5 - 15x4y + 90x3y2 - 270x2y3 + 405xy4 - 243y5 8. 5 4 xy 4 9. (a) 362,880 (b) 210 11.6 Exercises 1. permutation; combination 3. 6 5. 12 7. 24 9. 160 11. 5040 13. 3360 15. 132 17. 72 19. 5 21. 6 23. 1 25. 495 27. 1,860,480 29. 259,459,200 31. 15,504 33. 6435 35. (a) permutation (b) permutation (c) combination (d) combination (e) permutation (f) permutation (g) combination 37. 40 39. (a) 27,600 (b) 35,152 (c) 1104 41. 5040 43. (a) 17,576,000 (b) 17,576,000 (c) 456,976,000 45. 362,880 47. 120 49. 2730 51. (a) 120 (b) 6 53. 3,838,380 55. 12,650 57. (a) 15 (b) 10 59. 105; 1365 61. (a) 84 (b) 10 (c) 40 (d) 28 63. 1680 65. 15 67. 6,227,020,800 69. (a) 56 (b) 462 (c) 3080 (d) 8526 71. 1,000,000 73. 4096 75. 6 11.7 Exercises 1. 2; 1 2 3. 4; 1 4 5. 1 16 7. 51H, H2, 1H, T2, 1T, H2, 1T, T26 9. 51H, H, H2, 1H, H, T2, 1H, T, H2, 1T, H, H2, 1H, T, T2, 1T, H, T2, 1T, T, H2, 1T, T, T26 11. 511, 12, 11, 22, 11, 32, 12, 12, 12, 22, 12, 32, 13, 12, 13, 22, 13, 326 13. (a) 51H, H2, 1T, T26; 1 2 (b) 51H, H2, 1H, T2, 1T, H26; 3 4 15. (a) 51H, H, H2, 1T, T, T26; 1 4 (b) 51H, T, T2, 1T, H, T2, 1T, T, H2, 1T, T, T26; 1 2 17. (a) 511, 12, 12, 22, 13, 326; 1 3 (b) 511, 12, 11, 32, 12, 12, 12, 32, 13, 12, 13, 326; 2 3 (c) 512, 12, 12, 326; 2 9 19. 0.143 21. 3 to 7 23. (a) 1 4 (b) 3 4 (c) 1 2 (d) 11 26 (e) 1 to 3 25. (a) 0.521 (b) 0.895 (c) 0.403 (d) 6416 to 15,119 27. 1 28,561 ≈0.0000350 29. (a) 0.765 (b) 0.235 (c) 0.247 (d) 0.945 31. 0.3 33. 0.243 35. 0.036 37. 0.65 39. 0.35 41. (a) 0.048 (b) 0.977 (c) 0.103 (d) 0.897 43. (a) 0.404 (b) 0.047 (c) 0.002 Chapter 11 Review Exercises 1. 1 2 , 2 3 , 3 4 , 4 5 , 5 6 ; neither 3. 8, 10, 12, 14, 16; arithmetic 5. 5, 2, -1, -4, -7; arithmetic 7. 4, 4.5, 5, 5.5, 6 9. 3p - 2, 2p - 1, p, 1, -p + 2 11. -5, -1, - 1 5 , - 1 25 , - 1 125 13. {1; -8 A 1 2B n-1 = - A1 2B n-4 or -8 A - 1 2B n-1 = A - 1 2B n-4 15. -x + 61 17. 612 19. 4 25 21. -40 23. 1 25. 73 12 27. 3,126,250 29. 4 3 31. - 4 5 33. 36 35. diverges 37. -10
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