HMMT 二月 2009 · 冲刺赛 · 第 26 题

HMMT February 2009 — Guts Round — Problem 26

[ 12 ] Define the sequence { x } by x = 2009 and x = − x for all n ≥ 1. Compute the i i ≥ 0 0 n k k =0 n ∑ 2009 n value of 2 x . n k =0

解析

[ 12 ] Define the sequence { x } by x = 2009 and x = − x for all n ≥ 1. Compute the i i ≥ 0 0 n k k =0 n ∑ 2009 n value of 2 x . n n =0 Answer: 2009 Solution: We have nx ( n − 1) x n n − 1 − = x + x + ... + x = x + n − 1 n − 2 0 n − 1 2009 2009 n − 2010 n , which yields the recursion x = x . Unwinding this recursion, we find x = ( − 1) · 2009 · n n − 1 n n ( ) 2008 . Thus n ( ) 2009 2009 ∑ ∑ 2008 n n 2 x = ( − 2) · 2009 · n n k =0 k =0 ( ) 2008 ∑ 2008 n = 2009 ( − 2) n k =0 2008 = 2009( − 2 + 1) as desired.