返回题库

HMMT 十一月 2012 · 冲刺赛 · 第 26 题

HMMT November 2012 — Guts Round — Problem 26

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 13 ] x , x , x , . . . is a sequence of real numbers satisfying x = 1, x = 2, and x = 2 x − x + 2
    1 2 3 1 2 n +1 n n − 1
    for n ≥ 2. Find x .
    2012
    2 2
解析

英文解析

  1. [ 13 ]
    2013 n +1
    Answer: 2 − 6036 Let y = x − 2 . Note thatn nx = 2 x − x + 2 ⇔ y = 2 y − y ⇔ y − y = y − y .nn +1 n n − 1 n +1 n n − 1 n +1 n n n − 1
    Using the values for x , x , we get that y = − 3 and y = − 6, so y − y = ( − 6) − ( − 3) = − 3. By
    1 2 1 2 n +1 nn +1 2013
    induction, y = − 3 n . Then, we get that x = 2 − 3 n , so x = 2 − 6036.
    n n 2012
    Guts Round