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HMMT 十一月 2013 · 冲刺赛 · 第 8 题

HMMT November 2013 — Guts Round — Problem 8

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 7 ] Evaluate the expression 1 , 1 2 − 1 2 − 1 2 − · · · 1 2 − 2 where the digit 2 appears 2013 times. 2 2 2 2
解析
  1. [ 7 ] Evaluate the expression 1 , 1 2 − 1 2 − 1 2 − · · · 1 2 − 2 where the digit 2 appears 2013 times. 2013 Answer: Let f ( n ) denote the corresponding expression with the digit 2 appearing exactly n 2014 1 1 1 N times. 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 n 2013 f ( n ) = for all n ≥ 1, so f (2013) = . n +1 2014 2 2 2 2