返回题库

HMMT 十一月 2017 · 冲刺赛 · 第 7 题

HMMT November 2017 — Guts Round — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 3 ] Compute 2 2 2 2 2 2 2 2 2 2 2 2 100 + 99 − 98 − 97 + 96 + 95 − 94 − 93 + . . . + 4 + 3 − 2 − 1 .
解析
  1. [ 3 ] Compute 2 2 2 2 2 2 2 2 2 2 2 2 100 + 99 − 98 − 97 + 96 + 95 − 94 − 93 + . . . + 4 + 3 − 2 − 1 . Proposed by: Michael Tang Answer: 10100 2 2 2 2 2 Note that ( n + 3) − ( n + 2) − ( n + 1) + n = 4 for every n . Therefore, adding 0 to the end of the 2 given sum and applying this identity for every four consecutive terms after 100 , we see that the given 2 sum is equivalent to 100 + 25 · 4 = 10100 . 2 2 Alternatively, we can apply the difference-of-squares factorization to rewrite 100 − 98 = (100 − 2 2 98)(100 + 98) = 2(100 + 98), 99 − 97 = (99 − 97)(99 + 97) = 2(99 + 97), etc. Thus, the given sum is 100 · 101 equivalent to 2(100 + 99 + · · · + 2 + 1) = 2 · = 10100. 2