怎样计算1平方+2平方+3平方+4平方+.+n平方

Question

怎样计算1平方+2平方+3平方+4平方+.+n平方

Answer 1

n^2 = n(n+1) -n
=(1/3)[n(n+1)(n+2) -(n-1)n(n+1)] -(1/2)[n(n+1) -(n-1)n]
1^2+2^2+...+n^2
=(1/3)n(n+1)(n+2) -(1/2)n(n+1)
=(1/6)n(n+1)( 2(n+2)-3)
=(1/6)n(n+1)(2n+1)