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HMMT 二月 2004 · 代数 · 第 8 题

HMMT February 2004 — Algebra — Problem 8

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Let x be a real number such that x + 4 x = 8. Determine the value of x + 64 x . 2
解析
  1. Let x be a real number such that x + 4 x = 8. Determine the value of x + 64 x . Solution: 128 2 n +3 n +1 n For any integer n ≥ 0, the given implies x = − 4 x + 8 x , so we can rewrite any such power of x in terms of lower powers. Carrying out this process iteratively gives 7 5 4 x = − 4 x + 8 x 4 3 2 = 8 x + 16 x − 32 x 3 2 = 16 x − 64 x + 64 x 2 = − 64 x + 128 . Thus, our answer is 128. 2