求完全平方数前N项的和题目可以是这样的,设数列{An},A1=1^2,A2=2^2...An=n^2,求{An}前N项和Sn

Question

求完全平方数前N项的和
题目可以是这样的,设数列{An},A1=1^2,A2=2^2...An=n^2,求{An}前N项和Sn

Answer 1

(n+1)^3-n^3=3n^2+3n+1
n^3-(n-1)^3=3(n-1)^2+3(n-1)+1
……
2^3-1^1=3*1^2+3*1+1
全都加起来
左边正负何以抵消
(n+1)^3-1^1=3*[n^2+(n-1)^2+……+1^2]+3*[n+(n+1)+……+1]+1*n
n^2+(n-1)^2+……+1^2={(n+1)^3-1-3*[n+(n+1)+……+1]-n}/3
={(n+1)^3-1-3n(n+1)/2-n}/3
={(n+1)^3-3n(n+1)/2-(n+1)}/3
=(n+1)(n^2+2n+1-3n/2-1)/3
=(n+1)(n^2+n/2)/3
=(n+1)(2n^2+n)/6
=n(n+1)(2n+1)/6