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

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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

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• $f(n) = \frac{n(n+1)(2n+1)}{6}, f(n)-f(n-1) = n^2, f(0) = 0$ – reuns Mar 24 '17 at 11:08

## 2 Answers

Yes, there is a graphic proof by Man-Keung Siu, appeared in: Mathematics Magazine March, 1984

• This is very elegant. – mlc Mar 24 '17 at 11:07

The answer is a polynomial of the third degree because the first order finite difference of a polynomial is a polynomial of one degree lower.

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 $(3,14/3)$.