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HMMT 二月 2009 · GEN2 赛 · 第 7 题

HMMT February 2009 — GEN2 Round — Problem 7

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 5 ] Let F be the Fibonacci sequence, that is, F = 0, F = 1, and F = F + F . Computen 0 1 n +2 n +1 n
    ∑
    ∞
    F / 10 .nnn =0
解析

英文解析

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