因为（k+1）^3=k^3+3k^2+3k+1.1
k^3=(k-1)^3+3(k-1)^2+3(k-1)+1.2
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2^3=1^3+3*1^2+3*1+1.k
k式相加：(k+1)^3-1=3(k^2+.+1)+3(k+k-1+.+1)+k
所以3（k^2+...+1)
=(k+1)[(k+1)^2-1-k-(3k(k+1)/2)]
=k(k+1)(2k+1)
故1^2+2^2+3^2+...+N^2=N(N+1)(2N+1)/6
(n+1)^3-n^3=3n^2+3n+12^3-1^3=3*1^2+3*1+1
3^3-2^3=3*2^2+3*2+1
4^3-3^3=3*3^2+3*3+1
.(n+1)^3-n^3=3n^2+3n+1
以上各式相加,可得:(n+1)^3-1^3=3(
1^2+2^2+3^2+...+n^2)+3(1+2+3+4+5+6+.+n)+n
即n^3+3n^2+3n=3(1^2+2^2+3^2+...+n^2)+3n(n+1)/2+n
整理即可得1^2+2^2+3^2+...+N^2=N(N+1)(2N+1)/6