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HMMT 十一月 2013 · 冲刺赛 · 第 8 题

HMMT November 2013 — Guts Round — Problem 8

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 7 ] Evaluate the expression
    ,1
    2 −1
    2 −1
    2 − · · ·1
    2 −1
    where the digit 2 appears 2013 times.2
    2 2 2 2
解析

英文解析

  1. [ 7 ] Evaluate the expression
    ,1
    2 −1
    2 −1
    2 − · · ·1
    2 −1
    where the digit 2 appears 2013 times.2
    2013
    Answer: Let f ( n ) denote the corresponding expression with the digit 2 appearing exactly n
    2014
    1 1 1 Ntimes. Then f (1) = and for n > 1, f ( n ) = . By induction using the identity = ,
    N − 1
    2 2 − f ( n − 1) N +1
    2 −
    n 2013Nf ( n ) = for all n ≥ 1, so f (2013) = .
    n +1 2014
    2 2 2 2