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HMMT 二月 2004 · 冲刺赛 · 第 14 题

HMMT February 2004 — Guts Round — Problem 14

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [7] If a = 1 , a = 0, and a = a + for all n ≥ 1, compute a .
    1 2 n +1 n 2004 2
解析

英文解析

  1. If a = 1 , a = 0, and a = a + for all n ≥ 1, compute a .
    1 2 n +1 n 2004
    10022
    Solution: − 2
    By writing out the first few terms, we find that a = − 4 a . Indeed,
    n +4 na = 2( a − a ) = 2( a − 2 a ) = 2( − 2 a ) = − 4 a .
    n +4 n +3 n +2 n +2 n +1 n nk
    Then, by induction, we get a = ( − 4) for all positive integers k , and setting k = 501
    4 kgives the answer.