HMMT 二月 2004 · 代数 · 第 4 题

英文解析

4. Evaluate the sum
   1 1 1 1
   √ √ √ √

• • • · · · + .
      2 b 1 c + 1 2 b 2 c + 1 2 b 3 c + 1 2 b 100 c + 1
      Solution: 190 / 21
      The first three terms all equal 1 / 3, then the next five all equal 1 / 5; more generally,
      ⌊ ⌋ ⌊ ⌋
      √
      √
      2 2
      for each a = 1 , 2 , . . . , 9, the terms 1 / (2 a + 1) to 1 / (2 a + 2 a + 1) all equal
      1 / (2 a + 1), and there are 2 a + 1 such terms. Thus our terms can be arranged into
      ⌊ ⌋
      √
      9 groups, each with sum 1, and only the last term 1 / (2 100 + 1) remains, so theanswer is 9 + 1 / 21 = 190 / 21.
      − 1