homework 13 solution - Selected Answers to HW 13 December 4...

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Selected Answers to HW 13 December 4, 2012 Note: 1. There is a possibility of typos; in case you notice one, let me know. 2. These are just the answers; you must show work in your homework/quiz to get full credit. Section 8.2 4. a) a n = 3 5 ( - 2) n + 12 5 · 3 n . b) a n = 3 · 2 n - 5 n . c) a n = 3 · 2 n + 4 n . d) a n = 4 - 3 n . e) a n = 3( - 1) n + 2. g) a n = - ( - 5) n + 3. 8. a) L n = L n - 1 2 + L n - 2 2 . b) L n = 800000 3 ( - 1 2 ) n + 700000 3 . 12. a n = 6 - 2( - 1) n - 2 n . 14. a n = 1 + ( - 1) n + 2 n . 20. a n = α 1 · 2 n + α 2 n · 2 n + α 3 ( - 2) n + α 4 n ( - 2) n . 22. a n = ( α 1 , 1 + α 1 , 2 n + α 1 , 3 n 2 )( - 1) n + ( α 2 , 1 + α 2 , 2 n )2 n + ( α 3 , 1 + α 3 , 2 n )5 n + α 4 , 1 7 n . Section 8.4 2. 1 + 4 x + 16 x 2 + 64 x 3 + 256 x 4 = 1 - (4 x ) 5 1 - 4 x . 4. c) 3 x 2 1+ x . d) x + 1 1 - x . e) (1 + 2 x ) 7 . h) 1 1 - x 2 . 1
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6. a) - 1 1 - x . b) 2 x 1 - 2 x . c) 2 x - 1 (1 - x ) 2 . d) e x - 1 x . e) x 2 (1 - x ) 3 . f) (1+ x ) 10 - 1 x . 8. c) a n = 2 n 2 n even , 0 n odd . d) a n = ( n 2 ) . f) a n = ( - 1) n ( n +2 2 ) n < 3 , ( - 1) n ( n +2 2 ) + ( - 1) n +3 ( n - 1 2 ) n 3 . g) a n = 0 n = 3 k for some k Z , 1 n = 3 k + 1 for some k Z , - 1 n = 3 k + 2 for some k Z . 10. a) 10. b) 10. c) 3. d) 2. e) 0. 12. d) 4 12 ( 14 2 ) . e) ( - 4) 9
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Unformatted text preview: n is the sum of exactly k rolls of a die is the coeffecient of x n in the generating function ( x + x 2 + ··· + x 6 ) k . Now, we can achieve n as a sum in 0 rolls, 1 roll, 2 rolls, etc., i.e., when k = 0 , 1 , 2 ,... Therefore, the total number of ways in which we can obtain n as the sum is the coeffecient of x n in the generating function ∞ X k =0 ( x + x 2 + ··· + x 6 ) k = 1 1-x-x 2-···-x 6 . Section 8.5 8. 270-64-94-58 + 26 + 28 + 22-14 = 116. 12. 31 + 10-3 = 38. 2...
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  • Fall '12
  • Gupta
  • Math, Summation, Generating function

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