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PUMaC 2013 · 个人决赛（B 组） · 第 1 题

PUMaC 2013 — Individual Finals (Division B) — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
PUMaC

题目详情

  1. Let a = 2013 and a = 2013 for all positive integers n . Let b = 1 and b = 2013 1 n +1 1 n +1 for all positive integers n . Prove that a > b for all positive integers n . n n
解析
  1. Let a = 2013 and a = 2013 for all positive integers n . Let b = 1 and b = 2013 1 n +1 1 n +1 for all positive integers n . Prove that a > b for all positive integers n . n n Solution We claim that a ≥ 2013 b for all positive integers n . This is clearly true for n = 1. If n n a ≥ 2013 b for some positive integer k , then k k a 2013 b b 2012 b k k k k a = 2013 ≥ 2013 = 2013 · 2013 ≥ 2013 b . k +1 k +1