(n+1)³-n³=3n²+3n+1
n³-(n-1)³=3(n-1)²+3(n-1)+1
……
2³-1³=3×1²+3×1+1
相加,左边中间正负抵消
(n+1)³+1=3(1²+2²+……+n²)+3(1+2+……+n)+n
1+2+……+n=n(n+1)/2
代入整理得
1²+2²+……+n²=n(n+1)(2n+1)/6