PUMaC 2012 · 代数（B 组） · 第 2 题
PUMaC 2012 — Algebra (Division B) — Problem 2
题目详情
- [ 3 ] Define a sequence a such that a = a − a . Let a = 6 and a = 5. Find a . n n n − 1 n − 2 1 2 n n =1 √ √ √ √ 3 3
解析
- [ 3 ] Define a sequence a such that a = a − a . Let a = 6 and a = 5. Find a . n n n − 1 n − 2 1 2 n n =1 Solution: a = 6, a = 5, a = − 1, a = − 6, a = − 5, a = 1, a = 6, a = 5, so the sequence 1 2 3 4 5 6 7 8 starts to repeat itself. The sum of the first 6 terms is 0, so the sum of the first 996 terms is 0; thus, the sum of the first 1000 terms is 6 + 5 + ( − 1) + ( − 6) = 4. Answer: 4 √ √ √ √ 3 3