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HMMT 二月 2009 · GEN2 赛 · 第 7 题

HMMT February 2009 — GEN2 Round — Problem 7

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 5 ] Let F be the Fibonacci sequence, that is, F = 0, F = 1, and F = F + F . Compute n 0 1 n +2 n +1 n ∑ ∞ n F / 10 . n n =0
解析
  1. [ 5 ] Let F be the Fibonacci sequence, that is, F = 0, F = 1, and F = F + F . Compute n 0 1 n +2 n +1 n ∑ ∞ n F / 10 . n n =0 Answer: 10 / 89 ∑ ∞ n Solution: Write F ( x ) = F x . Then the Fibonacci recursion tells us that F ( x ) − xF ( x ) − n n =0 2 2 x F ( x ) = x , so F ( x ) = x/ (1 − x − x ) . Plugging in x = 1 / 10 gives the answer.