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HMMT 二月 2009 · 冲刺赛 · 第 26 题

HMMT February 2009 — Guts Round — Problem 26

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

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

英文解析

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