222a4s - MATH 222 FALL 2011, Assignment Four (Answer Key)...

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Unformatted text preview: MATH 222 FALL 2011, Assignment Four (Answer Key) 1. For n 1, let b n denote the number of ways to express n as the sum of 1s and 2s, taking order into account. Thus, b 4 = 5 because 4 = 1 + 1 + 1 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 2 = 1 + 2 + 1. (a) Find the first five terms of the sequence { b n } . b 1 = 1 , b 2 = 2 , b 3 = 3 , b 4 = 5 , b 5 = 8 (b) Find a recursive definition for b n . Divide all expressions S n of n as sum of 1s and 2s into two groups S 1 n and S 2 n where S 1 n consists of those whose last summand is 1 and S 2 n consists those whose last summand is 2. We have | S n | = b n , | S 1 n | = b n- 1 and | S 2 n | = b n- 2 for each n 3, so b n = b n- 1 + b n- 2 , i.e., the sequence { b n } satisfies the Fibonacci recurrence equation. 2. Find the sequence whose generating function is x 1- 2 x 2 . x 1- 2 x 2 = x i =0 (2 x 2 ) i = i =0 2 i x 2 i +1 = j =0 2 j- 3 2 [1- (- 1) j ] x j So the sequence is 2 j- 3 2 [1- (- 1) j ], j 0....
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222a4s - MATH 222 FALL 2011, Assignment Four (Answer Key)...

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