HMMT 十一月 2009 · 冲刺赛 · 第 23 题
HMMT November 2009 — Guts Round — Problem 23
题目详情
- [ 12 ] Let a , a , . . . be a sequence such that a = 3, a = 2, and a = a + a for all n ≥ 0. Find 0 1 0 1 n +2 n +1 n 8 ∑ a n . a a n +1 n +2 n =0
解析
- [ 12 ] Let a , a , . . . be a sequence such that a = 3, a = 2, and a = a + a for all n ≥ 0. Find 0 1 0 1 n +2 n +1 n 8 ∑ a n . a a n +1 n +2 n =0 a − a 105 a n +2 n +1 1 1 n Answer: We can re-write as = − . We can thus re-write the sum 212 a a a a a a n +1 n +2 n +1 n +2 n +1 n +2 as ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 1 105 − + − + − + . . . + − = − = − = . a a a a a a a a a a 2 212 212 1 2 2 3 4 3 9 10 1 10