Section 13.1 AN97 31. (I) = ⋅ = − n 1: 2 3 2 1 1 and − = 3 1 2 1 (II) If + + + + ⋅ = − − 2 6 18 2 3 3 1, k k 1 then ( ) + + + + ⋅ + ⋅ = + + + + ⋅ + ⋅ ( ) − + − − 2 6 18 2 3 2 3 2 6 18 2 3 2 3 k k k k 1 1 1 1 = − + ⋅ = ⋅ − = − + 3 1 2 3 3 3 1 3 1. k k k k 1 32. (I) ( ) = ⋅ − = n 1: 3 1 2 1 2 and [ ] ⋅ ⋅ ⋅ − ⋅ − = 1 2 1 6 1 3 1 1 1 2 (II) If ( ) ( ) + + + + − = − − k k k k 1 4 7 3 2 1 2 6 3 1 , 2 2 2 2 2 then ( ) ( ) [ ] + + + + − + + − k k 1 4 7 3 2 3 1 2 2 2 2 2 2 [ ] ( ) ( ) ( ) ( ) ( ) ( ) = + + + + − + + = −−++= −−+ ++ k k k k k k k k k k k 1 4 7 3 2 3 1 1 2 6 3 1 3 1 1 2 6 3 9 6 1 2 2 2 2 2 2 2 3 2 2 [ ] ( ) ( ) ( ) ( ) ( ) ( ) = + ++= + ++= + +−+− k k k k k k k k k 1 2 6 15 11 2 1 2 1 6 9 2 1 2 1 6 1 3 1 1 . 3 2 2 2 33. 10 34. + + + + + x x x x x 10 40 80 80 32 5 4 3 2 35. − + − + x x x x 81 432 864 768 256 4 3 2 36. 144 37. 84 38. (a) 8 bricks (b) 1100 bricks 39. 360 40. (a) ( ) = 20 3 4 135 16 ft 3 (b) ( ) 20 3 4 ft n (c) 13 times (d) 140 ft 41. $171,647.33 42. $58,492.93 43. (a) A A A 700, 1.00125 125 n n 0 1 = = + − (b) 12 months Chapter Test (page 907) 1. 0, 3 10 , 8 11 , 5 4 , 24 13 2. 4, 14, 44, 134, 404 3. − + = 2 3 4 4 9 61 36 4. − − − − = − 1 3 14 9 73 27 308 81 680 81 5. ∑ ( ) ( ) − + + = k k 1 1 4 k k 1 10 6. Neither 7. Geometric; ( ) = = − r S 4; 2 3 1 4 n n 8. Arithmetic: ( ) = − = − d S n n 8; 2 4 n 9. Arithmetic; ( ) = − = − d S n n 1 2 ; 4 27 n 10. Geometric; ( ) = = − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ r S 2 5 ; 125 3 1 2 5 n n 11. Neither 12. Converges; 1024 5 13. + + + + + m m m m m 243 810 1080 720 240 32 5 4 3 2 14. First we show that the statement holds for n 1. 1 1 1 1 1 2. ( ) = + = + = The equality is true for = n 1, so Condition I holds. Next we assume that k k 1 1 1 1 1 2 1 1 3 1 1 1 ( )( )( ) ( ) + + + + = + is true for some k, and we determine whether the formula then holds for + k 1. k k k k k k k k k k k 1 1 1 1 1 2 1 1 3 1 1 1 1 1 1 1 1 1 1 2 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 ( ) ( ) ( ) ( )( )( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) + + + + + + = + + + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + + = + + + = + ⋅ + + ⋅ + = + + = + Condition II also holds. So, the statement holds true for all natural numbers. 15. After 10 years, the car will be worth $6103.11. 16. The weightlifter will have lifted a total of 8000 pounds after 5 sets. Cumulative Review (page 908) 1. { } − − i i 3, 3, 3, 3 2. (a) y 24 x 6 (b) − + − + ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − − + − + ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ ⎧ ⎨ ⎪⎪ ⎩⎪⎪ ⎫ ⎬ ⎪⎪ ⎭⎪⎪ 1 3601 18 , 1 3601 6 , 1 3601 18 , 1 3601 6 (c) The circle and the parabola intersect at − + − + ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − − + − + ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ 1 3601 18 , 1 3601 6 , 1 3601 18 , 1 3601 6 3. { ( )} ln 5 2 4. = − y x5 10 5. ( ) ( ) + + − = x y 1 2 25 2 2 6. (a) 5 (b) 13 (c) + − x x 6 3 2 1 (d) x x 1 2 { } ≠ (e) − − x x 7 2 2 (f) { } ≠ x x 2 (g) ( ) ( ) = − −g x x 1 2 1 ; 1 all reals (h) { } ( ) = − ≠ −f x x x x x 2 3 ; 3 1 7. + = x y 7 16 1 2 2 8. ( ) ( ) + = − x y 1 4 2 2 9. θ ( ) = + − = r x y 8 sin ; 4 16 2 2 10. π { } 3 2 11. π2 3 12. (a) − 15 4 (b) − 15 15 (c) − 15 8 (d) 7 8 (e) + = + 1 15 4 2 4 15 2 2 CHAPTER 13 Counting and Probability 13.1 Assess Your Understanding (page 915) 5. subset; ⊆ 6. finite 7. T 8. b 9. {}{}{}{}{ }{ }{ }{ }{ }{ }{ } { } { } { } { } ∅ a b c d ab ac ad bc bd cd abc bcd acd abd abcd , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 11. 25 13. 40 15. 25 17. 37 19. 18 21. 5 23. 15 different arrangements 25. 9000 numbers 27. 175; 125 29. (a) 15 (b) 15 (c) 15 (d) 25 (e) 40 31. (a) 14.7 million (b) 76.8 million 33. 480 portfolios 36. y x 6 6 26 26 (x22) 2 1 (y11) 2 5 9 37. ≈ ° ≈ ° ≈ ° A B C 41.4 , 41.4 , 97.2 38. − 2, 5, 2 39. { } 1 18 40. { } −6 2, 0, 6 2 41. x y x y 3, 2; 8, 3or 3, 2 , 8,3 ( ) ( ) = = − = = − 42. − + − x x x 6 31 43 28 3 2 43. Converges; 10 44. (a) = x 15 8 (b) at = x 2 45. ( ) − + + + x x x 5 2 1 7 1 2
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