HMMT 二月 2006 · 代数 · 第 9 题
HMMT February 2006 — Algebra — Problem 9
题目详情
英文原题
- Compute the value of the infinite series
∞
4 2
∑
n + 3 n + 10 n + 10
n 4
2 · ( n + 4)
n =2
解析
英文解析
- Compute the value of the infinite series
∞
4 2
∑
n + 3 n + 10 n + 10
n 4
2 · ( n + 4)
n =2
Answer:113
Solution: We employ the difference of squares identity, uncovering the factorization 10
4 2 2 2 2 2
of the denominator: n + 4 = ( n + 2) − (2 n ) = ( n − 2 n + 2)( n + 2 n + 2). Now,
4 2 2
n + 3 n + 10 n + 10 3 n + 10 n + 6 = 1 +
4 4
n + 4 n + 4
4 1 = 1 + −
2 2
n − 2 n + 2 n + 2 n + 2
∞ ∞
4 2
∑ ∑
n + 3 n + 10 n + 10 1 4 1 = ⇒ = + −
n 4 n n 2 n 2
2 · ( n + 4) 2 2 · ( n − 2 n + 2) 2 · ( n + 2 n + 2)
n =2 n =2
∞
∑
1 1 1 = + −
n − 2 2 n 2
2 2 · (( n − 1) + 1) 2 · (( n + 1) + 1)
n =2
1 1 1 1 1 11
The last series telescopes to + , which leads to an answer of + + = .
2 10 2 2 10 10