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HMMT 二月 2010 · TEAM2 赛 · 第 5 题

HMMT February 2010 — TEAM2 Round — Problem 5

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 25 ] Compute 98 ∑ 2 1 √ + √ √ . √ n + n + 2 n + 1 + n + 2 n =1
解析
  1. [ 25 ] Compute 98 ∑ 2 1 √ √ √ √ + . n + n + 2 n + 1 + n + 2 n =1 √ √ Answer: 3 11 − 2 2 + 19 Rationalizing the denominator of both terms in the summation yields 98 ∑ √ √ √ √ √ √ √ √ √ √ n + 2 − n + n + 2 − n + 1 = 2 n + 2 − ( n + n + 1). Then the sum 2 n + 2 − ( n + n + 1) n =1 √ √ √ √ √ √ √ √ telescopes. All terms cancel except for − ( 1 + 2) − 2 + 2 99 + 2 100 − 99 = 3 11 − 2 2 + 19.