222a5s - MATH 222 FALL 2011, Assignment Five (Answer Key)...

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(Answer Key) 1. Use generating functions to solve the following recurrence relations with initial con- ditions. (a) a k - a k - 1 = 2 k + 1 , k 1 , a 0 = 1; Let f ( x ) = k =0 a k x k . Then a k x k - a k - 1 x k = (2 k + 1) x k ( k 1) k =1 a k x k - k =1 a k - 1 x k = k =1 (2 k + 1) x k f ( x ) - a 0 - xf ( x ) = k =1 (2 k + 2) x k - k =1 x k (1 - x ) f ( x ) - 1 = 2 (1 - x ) 2 - 2 - 1 1 - x + 1 (1 - x ) f ( x ) = 2 (1 - x ) 2 - 1 1 - x f ( x ) = 2 (1 - x ) 3 - 1 (1 - x ) 2 = k =0 2 ( k +2 k ) x k - k =0 ( k +1 k ) x k = k =0 ( k + 1) 2 x k Consequently, a k = ( k + 1) 2 for all k 0. (b) a k - 2 a k - 1 = 2 k , k 1 , a 0 = 1. Let f ( x ) = k =0 a k x k . Then a k x k - 2 a k - 1 x k = 2 k x k ( k 1) k =1 a k x k - k =1 2 a k - 1 x k = k =1 2 k x k f ( x ) - a 0 - 2 xf ( x ) = 2 x/ (1 - 2 x ) (1 - 2 x ) f ( x ) = a 0 + 2 x/ (1 - 2 x ) = 1 / (1 - 2 x ) f ( x ) = 1 / (1 - 2 x
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222a5s - MATH 222 FALL 2011, Assignment Five (Answer Key)...

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