HMMT 十一月 2012 · 冲刺赛 · 第 26 题
HMMT November 2012 — Guts Round — Problem 26
题目详情
- [ 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
- [ 13 ] 2013 n +1 Answer: 2 − 6036 Let y = x − 2 . Note that n n n x = 2 x − x + 2 ⇔ y = 2 y − y ⇔ y − y = y − y . n +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 n n +1 2013 induction, y = − 3 n . Then, we get that x = 2 − 3 n , so x = 2 − 6036. n n 2012 Guts Round