- 1的平方加２的平方一直加到n的平方的公式怎么来得？就问公式的来源
- 就问公式的来源？
- formulla:(n+1)^2-n^3=3n^2+3n+1.......(1)
--->n^3-(n-1)^3=3(n-1)^2+3(n-1)+1......(2)
--->(n-1)^3-(n-2)^3=3(n-2)^3+3(n-2)+1......(3)
...............
3^3-2^3=3*2^2+3*2+1......(n-1)
2^3-1^3=3*1^2+3*1+1......(n)
(1)+(2)+(3)+......+(1):
(n+1)^3-1^3=3(1^2+2^2+3^3+......+n^2)+3(1+2+3+......+1)+(1+1+......+1)
--->n^3+3n^2+3n=3(1+2+3+......+n^2)+3*n(n+1)/2+n
--->3(1^2+2^2+3^2+......+n^2)=n^3+3n^2+3n-3n(n+1)/2-n
=n^3+3n^2/2+n/2
--->1^2+2^2+......+n^2=(2n^3+3n^2+n)/6
=n(2n^2+3n+1)/6
=n(n+1)(2n+1)/6.