HMMT 二月 2004 · 代数 · 第 8 题

英文解析

8. Let x be a real number such that x + 4 x = 8. Determine the value of x + 64 x .
   Solution: 128
   n +3 n +1 n 2
   For any integer n ≥ 0, the given implies x = − 4 x + 8 x , so we can rewrite anysuch 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 = − 64 x + 128 .2
   Thus, our answer is 128. 2