AN94 Answers: Chapter 12 67. 210 beige and 190 blue 69. T n T 5.5 67 ; 39.5 F n 5 { } { } = − + = ° 71. 14 75. 16.42% 76. = − v i j 4 6 77. Ellipse: Center: ( ) 0, 0 ; Vertices: 0, 5 , 0, 5 ; ( ) ( ) − Foci: 0, 21 , ( ) − 0, 21 , ( ) y x 4 6 2 6 22 24 26 26 (0, 25) (22, 0) (0, Ï21) (0, 2Ï21) (0, 5) (2, 0) 78. − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 1 2 0 3 2 1 79. − − − + + x x x x 1 1 1 1 2 80. 2 2 81. [ ] −3, 11 82. + − + 0, 1 2, 1 2 83. Hyperbola 84. { } − 17 8 Historical Problems (page 883) 1. 1 2 3 loaves, 10 5 6 loaves, 20 loaves, 29 1 6 loaves, 38 1 3 loaves 2. (a) 1 person (b) 2401 kittens (c) 2800 12.3 Assess Your Understanding (page 883) 2. = − a a r n n 1 1 3. Geometric 4. − a r 1 5. b 6. T 7. F 8. T 9. = = = = = r s s s s 4; 4, 16, 64, 256 1 2 3 4 11. = = − = − = − = − r a a a a 1 2 ; 3 2 , 3 4 , 3 8 , 3 16 1 2 3 4 13. = = = = = r c c c c 2; 1 4 , 1 2 , 1, 2 1 2 3 4 15. = = = = = r e e e e 7 ; 7 , 7 , 7 , 7 1 4 1 1 4 2 1 2 3 3 4 4 17. = = = = = r t t t t 3 2 ; 1 2 , 3 4 , 9 8 , 27 16 1 2 3 4 19. = = ⋅ − a a 162; 2 3 n n 5 1 21. ( ) = = ⋅ − − a a 5; 5 1 n n 5 1 23. = = a a 0; 0 n 5 25. ( ) = = a a 9 3; 3 n n 5 27. = a 1 64 7 29. = a 1 15 31. = a 0.00000004 8 33. = ⋅ − a 6 3 n n 1 or = ⋅ a 2 3 n n 35. ( ) ( ) = − ⋅ − = − − − a 3 1 3 1 3 n n n 1 2 37. ( ) = − − − a 3 n n 1 39. = ⋅ = ⋅ − − a 7 15 15 7 15 n n n 1 2 41. ( ) − − 1 4 1 2n 43. ( ) − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 1 2 3 n 45. −1 2n 47. 49. 51. 53. Converges; 3 2 55. Converges; 16 57. Converges; 8 5 59. Diverges 61. Converges; 20 3 63. Diverges 65. Converges; 18 5 67. Converges; 6 69. Arithmetic; = d 1; 1375 71. Neither 73. Arithmetic; = − − d 2 3 ; 700 75. Neither 77. Geometric; ( ) = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ r 2 3 ; 2 1 2 3 50 79. Geometric; [ ] ( ) = − − − − r 2; 1 3 1 2 50 81. Geometric; ( )( ) = − + − r 3 ; 3 2 1 3 1 3 1 2 25 83. −4 85. $47,271.37 87. (a) 0.775 ft (b) 8th (c) 15.88 ft (d) 20 ft 89. $149,035.94 91. $68,059.40 93. $348.60 95. × 1.845 1019 97. 10 99. $72.67 per share 101. December 20; $9999.92 103. 18 105. Option A results in a higher salary in 5 years ($50,499 versus $49,522); option B results in a higher 5-year total ($225,484 versus $233,602). 107. Option 2 results in the most: $16,038,304; option 1 results in the least: $14,700,000. 109. Yes. A constant sequence is both arithmetic and geometric. For example, 3, 3, 3, . . . is an arithmetic sequence with = a 3 1 and = d 0 and is a geometric sequence with = a 3 and = r 1. 113. 2.121 114. −i j 8 17 15 17 115. − = x y 4 12 1 2 2 116. 54 117. 7.3 feet 118. ( ) ( )( )( ) =− + − − f x x x x 1 8 2 1 4 2 119. − − t 16 13 120. = − y x 5 121. ( ) = + g x x 7 5 122. ( )( )( )( ) − + − + x x x x 5 5 2 2 12.4 Assess Your Understanding (page 893) 1. a 2. infinite series 3. converges 4. T 5. converges; 2 7. diverges 9. converges; 1 2 11. converges; 0 13. diverges 15. converges; 0 17. converges; 3 19. converges; 4 21. converges; 1 23. diverges 25. diverges 27. diverges 29. 1, 4 3 , 13 9 , 40 27 , 121 81 31. 1, 3, 6, 10, 15 12.5 Assess Your Understanding (page 896) 1. (I) = ⋅ = n 1: 2 1 2 and ( ) + = 1 1 1 2 (II) If ( ) + + + + = + k k k 2 4 6 2 1 , then ( ) ( ) ( ) ++++ + +=++++ + + k k k k 2 4 6 2 2 1 2 4 6 2 2 1 ( ) ( ) ( )( ) ( ) ( ) [ ] = ++ +=++=+ +=+ ++ k k k k k k k k k 1 2 1 3 2 1 2 1 1 1 . 2 3. (I) = + = n 1: 1 2 3 and ( ) ⋅ ⋅ + = ⋅ = 1 2 1 1 5 1 2 6 3 (II) If ( ) ( ) + + + + + = + k k k 3 4 5 2 1 2 5 , then ( ) ( ) [ ] + + + + + + + + k k 3 4 5 2 1 2 ( ) ( ) [ ] ( ) ( ) ( )( ) ( ) ( ) [ ] =+++++ ++= +++= ++= + += + ++ k k k k k k k k k k k 3 4 5 2 3 1 2 5 3 1 2 7 6 1 2 1 6 1 2 1 1 5 . 2 5. (I) n 1: 3 1 1 2 ( ) = − = and 1 2 1 3 1 1 1 2 4 2 [ ] ⋅ ⋅ ⋅ + = ⋅ = (II) If ( ) ( ) + + + + − = + k k k 2 5 8 3 1 1 2 3 1 , then ( ) ( ) [ ] + + + + − + + − k k 2 5 8 3 1 3 1 1 ( ) ( ) [ ] ( ) ( ) ( ) ( )( ) = + + + + − + + = + + + = + + = + + k k k k k k k k k 2 5 8 3 1 3 2 1 2 3 1 3 2 1 2 3 7 4 1 2 1 3 4 2 ( ) ( ) [ ] = + + + k k 1 2 1 3 1 1 .
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