PUMaC 2017 · 代数（A 组） · 第 7 题

PUMaC 2017 — Algebra (Division A) — Problem 7

The sum ∞ k ∑ 2 k 2 5 + 1 k =0 p can be written in the form where p and q are relatively prime positive integers. Find p + q . q

解析

Let Ξ = . Note that k k =0 2 5 +1 ∞ k k k ∑ 2 2 1 2 n = = ( − 1) . k k k k 2 2 − 2 ( n +1)2 5 + 1 5 1 + 5 5 n =0 Therefore, we have ∞ ∞ ∞ k ∑ ∑ ∑ ∑ 2 1 n k Ξ = = ( − 1) 2 . k N ( n +1)2 5 5 k n =0 k =0 N =1 N =( n +1)2 m Given any N ∈ N , let N = a 2 where a and m are non-negative integers and a is odd. Then, for fixed N , we have m m − 1 ∑ ∑ ∑ m − k n k a 2 − 1 k m k ( − 1) 2 = ( − 1) 2 = 2 − 2 = 1 . k k =0 k =0 N =( n +1)2 Therefore, ∞ ∑ 1 1 Ξ = = , N 5 4 N =1 giving us an answer of 5 . Problem written by Matt Tyler