AN96 Answers: Chapter 12 31. (I) ( ) [ ] = + − = n a d a 1: 1 1 and ( ) ⋅ + ⋅ − = a d a 1 1 1 1 2 . (II) If ( ) ( ) ( ) [ ] ( ) ++++ +++− =+ − a a d a d a k d ka d k k 2 1 1 2 , then ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ++++ +++− ++ +− =+ − + + a a d a d a k d a k d ka d k k a kd 2 1 1 1 1 2 ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) [ ] = + + − + = + + + = + + + + − k a d k k k k a d k k k a d k k 1 1 2 2 1 1 2 1 1 1 1 2 . 33. (I) = n 3: The sum of the angles of a triangle is ( ) − ⋅ ° = ° 3 2 180 180 . (II) Assume that for some ≥ k 3, the sum of the angles of a convex polygon of k sides is ( ) − ⋅ ° k 2 180 . A convex polygon of + k 1 sides consists of a convex polygon of k sides plus a triangle (see the figure). The sum of the angles is ( ) ( ) ( ) [ ] − ⋅ ° + ° = − ⋅ ° = + − ⋅ ° k k k 2 180 180 1 180 1 2 180 . k sides k 1 1 sides 35. (a) 7; 15 (b) = − c 2 1 n n (c) (I) = n 1: one fold results in 1 crease and = − = c 2 1 1. 1 1 (II) If = − c 2 1, k k then ( ) = += −+= −+= − + + + c c2 1 2 2 1 1 2 2 1 2 1 k k k k k 1 1 1 (d) Each fold doubles the thickness so the stack thickness will be ⋅ = 2 0.02 mm 671,088.64 mm 25 (or about 671 meters). 37. { } 251 38. ( ) = = − − x y 1 2 , 3; 1 2 , 3 39. Left: 448.3 kg; right: 366.0 kg 40. − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 7 3 7 8 41. + + − x x 2 2 1 1 42. 61.4° 43. { } + ≈ ln 4 7 3 2.795 44. 39 13 45. = x 1 12.6 Assess Your Understanding (page 903) 1. Pascal triangle 2. 1; n 3. F 4. Binomial Theorem 5. 10 7. 21 9. 50 11. 1 13. ≈ × 1.8664 1015 15. ≈ × 1.4834 1013 17. + + + + + x x x x x 5 10 10 5 1 5 4 3 2 19. − + − + − + x x x x x x 12 60 160 240 192 64 6 5 4 3 2 21. + + + + x x x x 81 108 54 12 1 4 3 2 23. + + + + + x x y x y x y x y y 5 10 10 5 10 8 2 6 4 4 6 2 8 10 25. + + + + + + x x x x x x 6 2 30 40 2 60 24 2 8 3 5 2 2 3 2 1 2 27. + + + + + a x a bx y a b x y a b x y ab xy b y 5 10 10 5 5 5 4 4 3 2 3 2 2 3 2 3 4 4 5 5 29. 17,010 31. −101,376 33. 41,472 35. x 2835 3 37. x 314,928 7 39. 495 41. 3360 43. 1.00501 45. ( ) ( ) [ ] ( ) ( ) ( ) ( ) − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ = − − − = − = ⋅ − − = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ = − = = = n n n n n n n n n n n n n n n n n n n n n n 1 ! 1 ! 1 ! ! 1! 1! 1 ! 1 ! ; ! ! ! ! !0! ! ! 1 47. ( ) ( ) ( ) = + = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − n n n n n n n n 2 1 1 0 1 1 1 1 1 0 1 n n n n n 1 49. 1 51. 165 53. { } { } − ≈ ln 5 ln 6 ln 5 8.827 54. (a) 0 (b) 90° (c) Orthogonal 55. x y z 1, 3, 2; 1, 3, 2 ( ) = = = − − 56. Bounded y x (5, 0) (4, 2) (0, 0) (0, 6) 57. g f x x 4; Domain: , 2 2, 2 ( ) ( ) ( ] [ ) = − −∞ − ∪ ∞ 58. = − C 46 59. θ θ θ θ θ θ θ θ θ θ ( ) + = + = = ⋅ = sin sin tan sin 1 tan sin sec sin 1 cos tan 2 2 2 2 2 2 2 2 2 2 60. ( ) − + x x x 1 8 3 1 3 2/3 3 2 61. = = − x x 3, 1 62. f 2 5; 2, 5 ( ) ( ) − = − Review Exercises (page 906) 1. = − = = − = = − a a a a a 4 3 , 5 4 , 6 5 , 7 6 , 8 7 1 2 3 4 5 2. = = = = = c c c c c 2, 1, 8 9 , 1, 32 25 1 2 3 4 5 3. = = = = = a a a a a 3, 2, 4 3 , 8 9 , 16 27 1 2 3 4 5 4. = = = = = a a a a a 2, 0, 2, 0, 2 1 2 3 4 5 5. + + + = 6 10 14 18 48 6. ∑( ) − = + k 1 1 k k 1 13 1 7. Arithmetic; ( ) = = + d S n n 1; 2 11 n 8. Neither 9. Geometric; ( ) = = − r S 8; 8 7 8 1 n n 10. Arithmetic; ( ) = = − d S n n 4; 2 1 n 11. Geometric; ( ) = = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ r S 1 2 ; 6 1 1 2 n n 12. Neither 13. 9515 14. −1320 15. ≈ 1093 2187 0.49977 16. 682 17. 35 18. 1 1010 19. 9 2 20. { } { } = − a n5 4 n 21. { } { } = − a n 10 n 22. Converges; 9 2 23. Converges; 4 3 24. Diverges 25. Converges; 8 26. Converges; 7 27. Diverges 28. See page 890 29. See page 891 30. (I) = ⋅ = n 1: 3 1 3 and ( ) ⋅ + = 3 1 2 1 1 3 (II) If k k k 369 3 3 2 1 , ( ) + + + + = + then ( ) ( ) ( ) +++++ +=++++ + + k k k k 3 6 9 3 3 1 3 6 9 3 3 3 ( ) ( ) ( ) ( )( ) ( ) ( ) [ ] = ++ += +++= ++= + += + + + k k k k k k k k k k k k 3 2 1 3 3 3 2 3 2 6 2 6 2 3 2 3 2 3 2 1 2 3 1 2 1 1 . 2 2
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