HMMT 二月 2018 · 冲刺赛 · 第 11 题

HMMT February 2018 — Guts Round — Problem 11

解析

[ 7 ] FInd the value of 60 k 2 ∑ ∑ n . 61 − 2 n n =1 k =1 Proposed by: Henrik Boecken Answer: − 18910 Change the order of summation and simplify the inner sum: 60 k 60 60 2 2 ∑ ∑ ∑ ∑ n n = 61 − 2 n 61 − 2 n n =1 n =1 k =1 k = n 60 2 ∑ n (61 − n ) = 61 − 2 n n =1 Then, we rearrange the sum to add the terms corresponding to n and 61 − n : ( ) 60 30 2 2 2 ∑ ∑ n (61 − n ) n (61 − n ) (61 − n ) (61 − (61 − n )) = + 61 − 2 n 61 − 2 n 61 − 2(61 − n ) n =1 n =1 30 2 2 ∑ n (61 − n ) − n (61 − n ) = 61 − 2 n n =1 30 ∑ n (61 − n )( n − (61 − n )) = 61 − 2 n n =1 30 ∑ = − n (61 − n ) n =1 30 ∑ 2 = n − 61 n n =1 Finally, using the formulas for the sum of the first k squares and sum of the first k positive integers, we conclude that this last sum is 30(31)(61) 30(31) − 61 = − 18910 6 2 So, the original sum evaluates to − 18910.