Section 12.2 AN93 33. ( ) = − + a 1 n n 1 35. ( ) = − + a n 1 n n 1 37. = = = = = a a a a a 2, 5, 8, 11, 14 1 2 3 4 5 39. = − = = = = a a a a a 2, 0, 3, 7, 12 1 2 3 4 5 41. = = = = = a a a a a 5, 10, 20, 40, 80 1 2 3 4 5 43. = = = = = a a a a a 3, 3 2 , 1 2 , 1 8 , 1 40 1 2 3 4 5 45. = = = = = a a a a a 1, 2, 2, 4, 8 1 2 3 4 5 47. = = + = + = + = + a Aa Ada Ada Ada Ad , , 2 , 3 , 4 1 2 3 4 5 49. = = + = + + = + + + = + + + + a a a a a 2, 2 2, 2 2 2, 2 2 2 2 , 2 2 2 2 2 1 2 3 4 5 51. ( ) + + + +n 3 4 2 53. + + + + n 1 2 2 9 2 2 2 55. + + + + 1 1 3 1 9 1 3n 57. + + + 1 3 1 9 1 3n 59. ( ) − + − + − n ln 2 ln 3 ln 4 1 ln n 61. ∑ = k k 1 20 63. k k 1 k 1 13 ∑ + = 65. ∑ ( ) ( ) − = 1 1 3 k k k 0 6 67. ∑ = k 3 k n k 1 69. ∑( ) + = a kd k n 0 or ∑ ( ) [ ] + − = + a k d 1 k n 1 1 71. 200 73. 820 75. 1110 77. 1560 79. 3570 81. 44,000 83. (a) 2162 (b) After 26 months 85. (a) ( ) = = + − A A A 1500, 1.0125 750 n n 0 1 (b) After 99 quarters (c) $213,073.11 87. (a) $2930 (b) 14 payments have been made. (c) 36 payments; $3584.62 (d) $584.62 89. (a) ( ) = = − − a a a 150,000, 1.005 899.33 n n 0 1 (b) $149,850.67 (c) (d) After 58 payments or 4 years and 10 months later (e) After 359 payments of $899.33, plus last payment of $895.10 (f) $173,754.57 (g) (a) ( ) = = − − a a a 150,000, 1.005 999.33 n n 0 1 (b) $149,750.67 (c) (d) After 37 payments or 3 years and 1 month later (e) After 278 payments of $999.33, plus last payment of ( ) = $353.69 1.005 $355.46 (f) $128,169.20 91. 21 pairs 93. Fibonacci sequence 95. (a) 3.630170833 (b) 3.669060828 (c) 3.669296668 (d) 12 97. (a) = = = = = = = = a a a a a a a a 0.4; 0.7; 1; 1.6; 2.8; 5.2; 10; 19.6 1 2 3 4 5 6 7 8 (b) Except for term 5, which has no match, Bode’s formula provides excellent approximations for the mean distances of the planets from the sun. (c) The mean distance of Ceres from the sun is approximated by = a 2.8, 5 and that of Uranus is = a 19.6 8 . (d) = = a a 38.8; 77.2 9 10 (e) Pluto’s distance is approximated by a9, but no term approximates Neptune’s mean distance from the sun. (f) According to Bode’s Law, the mean orbital distance of Eris will be 154 AU from the sun. 99. (a) = ⋅ a I 0.95 n n 0 (b) 77 101. = = a a 2; 2.236067977; 2.236067977 0 5 103. = = a a 4; 4.582575695; 4.582575695 0 5 105. 1, 3, 6, 10, 15, 21, 28 107. ∑ ( ) = + + + + = = + = u n k n n 1 2 3 1 2 , n k n 1 and from Problem 106, ( )( ) = + + + u n n 1 2 2 . n 1 ( )( ) ( ) ( ) ( ) [ ] ( ) + = + + + + = + + + = + + u u n n n n n n n n 1 2 2 1 2 1 2 2 1 n n 1 2 111. $2654.39 112. ( ) ° + ° i 2 cos225 sin225 113. 0 114. ( ) ( ) − = + y x 4 16 3 2 115. = y 0 116. = ° = ° = ° A B C 23 , 42 , 115 117. 2 10 5 118. −2, 2 5 119. −2, 6 12.2 Assess Your Understanding (page 873) 1. arithmetic 2. F 3. 17 4. T 5. d 6. c 7. ( ) ( ) [ ] ( ) − =+− −+=+−+=+−−= − s s n n n n n n 4 1 4 4 3 4 3 1, n n 1 a constant; = = = = = d s s s s 1; 5, 6, 7, 8 1 2 3 4 9. ( ) ( ) [ ] ( ) ( ) − = −− −−=−− −−=−− −=−−+= − a a n n n n n n n n 2 5 2 1525225252725272, n n 1 a constant; = = − = − = = d a a a a 2; 3, 1, 1, 3 1 2 3 4 11. ( ) ( ) [ ] ( ) ( ) − =− −− − =−−−+=−−− =−−+=− − c c n n n n n n n n 6 2 6 2 1 6 2 6 2 2 6 2 8 2 6 2 8 2 2, n n 1 a constant; = − = = = = − d c c c c 2; 4, 2, 0, 2 1 2 3 4 13. t t n n n n n n n n 1 2 1 3 1 2 1 3 1 1 2 1 3 1 2 1 3 1 3 1 2 1 3 5 6 1 3 1 2 1 3 5 6 1 3 1 3 , n n 1 ( ) ( ) ( ) ( ) − = − − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ =−−−+=−−− =−−+ =− − a constant; = − = = − = − = − d t t t t 1 3 ; 1 6 , 1 6 , 1 2 , 5 6 1 2 3 4 15. ( ) ( ) −=− =−− =− −=−+= − − s s n n n n n n ln 3 ln 3 ln 3 1 ln 3 ln 3 ln 3 ln 3 ln 3 ln 3 ln 3 ln 3, n n n n 1 1 a constant; = = = = = d s s s s ln 3; ln 3, 2 ln 3, 3 ln 3, 4 ln 3 1 2 3 4 17. = − = a n a 3 1; 152 n 51 19. = − = − a n a 15 7 ; 342 n 51 21. ( ) = − = a n a 1 2 1 ; 25 n 51 23. = = a n a 2 ; 51 2 n 51 25. 200 27. −531 29. 83 2 31. = − = = + = − + − a d a a a n 13; 3; 3; 16 3 n n n 1 1 33. = − = = + = − + − a d a a a n 53; 6; 6; 59 6 n n n 1 1 35. = = − = − = − − a d a a a n 28; 2; 2; 30 2 n n n 1 1 37. = = − = − = − − a d a a a n 25; 2; 2; 27 2 n n n 1 1 39. n2 41. ( ) + n n 2 9 5 43. 1260 45. 195 47. −9312 49. 10,036 51. 12,240 53. −1925 55. 15,960 57. − 3 2 59. 24 terms 61. 1185 seats 63. 2160 seats 65. 8 yr
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