返回题库

HMMT 二月 2006 · 代数 · 第 4 题

HMMT February 2006 — Algebra — Problem 4

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Let a , a , . . . be a sequence defined by a = a = 1 and a = a + a for n ≥ 1. Find 1 2 1 2 n +2 n +1 n ∞ ∑ a n . n +1 4 n =1
解析
  1. Let a , a , . . . be a sequence defined by a = a = 1 and a = a + a for n ≥ 1. 1 2 1 2 n +2 n +1 n Find ∞ ∑ a n . n +1 4 n =1 1 Answer: 11 Solution: Let X denote the desired sum. Note that 1 1 2 3 5 X = + + + + + . . . 2 3 4 5 6 4 4 4 4 4 1 1 2 3 5 8 4 X = + + + + + + . . . 1 2 3 4 5 6 4 4 4 4 4 4 1 1 2 3 5 8 13 16 X = + + + + + + + . . . 0 1 2 3 4 5 6 4 4 4 4 4 4 4 so that X + 4 X = 16 X − 1, and X = 1 / 11.