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HMMT 二月 2002 · 冲刺赛 · 第 6 题

HMMT February 2002 — Guts Round — Problem 6

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ ± 6] Find all values of x that satisfy x = 1 − x + x − x + x − x + · · · (be careful; thisis tricky).
解析

英文解析

  1. Find all values of x that satisfy x = 1 − x + x − x + x − x + · · · (be careful; thisis tricky).
    √
    − 1 ± 5
    Solution: Multiplying both sides by 1 + x gives (1 + x ) x = 1, or x = . However,
    √2 − 1+ 5
    the series only converges for | x | < 1, so only the answer x = makes sense. 2