# How to sum $\sum_{k=1}^{n}n^2$ WITHOUT induction? [duplicate]

Is there any way to prove that: $$1^2+2^2+3^2+\ldots +n^2=\frac{n(n+1)(2n+1)}{6}$$ but WITHOUT using mathematical induction? (I don't know, maybe through some creative graphical demonstration?)

## marked as duplicate by PSPACEhard, Dietrich Burde, Juniven, projectilemotion, Simply Beautiful ArtMar 24 '17 at 11:21

• $f(n) = \frac{n(n+1)(2n+1)}{6}, f(n)-f(n-1) = n^2, f(0) = 0$ – reuns Mar 24 '17 at 11:08
This polynomial is obtained by Lagrangian interpolation on the points $(0,0),(1,1),(2,5)$ and $(3,14)$. As there is no independent term, you make is slightly easier by interpolating $P(n)/n$ on $(1,1),(2,5/2)$ and $(3,14/3)$.