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HMMT 二月 2000 · POW 赛 · 第 9 题

HMMT February 2000 — POW Round — Problem 9

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Ev aluate the sum + + + : : : + . 0 2 1 3 2 101 100
解析
  1. Let S b e the sum desired. Then 101 S = + + + : : : + 1 0 2 1 3 2 101 100 100 101 . No w, onsider a general term in this expression - i.e. . This is equal to 100 i +1 i 101 101 100! 101 100! 101! = = = . So, w e an simplify the terms i +1 (100 i )! i ! (101 ( i +1) )! i ! ( i +1) (101 ( i +1) )! i +1! i +1 101 101 101 101 101 101 to get 101 S = + + + : : : + + . Th us, 101 S = 2 1, 1 2 3 100 101 101 2 1 so S = . 101 n + m 1 n + m 1