由1²+2²+3²+.+n²=n(n+1)(2n+1)/6
∵(a+1)³-a³=3a²+3a+1(即(a+1)³=a³+3a²+3a+1)
a=1时:2³-1³=3×1²+3×1+1
a=2时:3³-2³=3×2²+3×2+1
a=3时:4³-3³=3×3²+3×3+1
a=4时:5³-4³=3×4²+3×4+1
.
a=n时:(n+1)³-n³=3×n²+3×n+1
等式两边相加:
(n+1)³-1=3(1²+2²+3²+.+n²)+3(1+2+3+.+n)+(1+1+1+.+1)
3(1²+2²+3²+.+n²)=(n+1)³-1-3(1+2+3+.+n)-(1+1+1+.+1)
3(1²+2²+3²+.+n²)=(n+1)³-1-3(1+n)×n÷2-n
6(1²+2²+3²+.+n²)=2(n+1)³-3n(1+n)-2(n+1)
=(n+1)[2(n+1)²-3n-2]
=(n+1)[2(n+1)-1][(n+1)-1]
=n(n+1)(2n+1)
∴1²+2²+.+n²=n(n+1)(2n+1)/6.