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Using induction it is straight forward to show $$\left(\sum_{k=1}^n k\right)^2=\sum_{k=1}^n k^3.$$ But is there also a geometrical interpretation that "proves" this fact? By just looking at those formulas I don't see why they should be equal.

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The clearest proof I've seen is this one, which comes from here: you just have to stare at it for a few seconds to see how it works. (It's a variant of the other proofs, of course, but actually has cubes in it.)

Animation

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There is this picture. Here, they represent $x^3$ as $x$ squares of side length $x$. The big square is the sum of all numbers up to $x$.

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    $\begingroup$ Who are they? Add a reference. $\endgroup$ – lhf May 30 '15 at 10:13
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page 86 ,85
book :proof without words
author : roger nelsen
you can find here many kind of this proof enter image description here

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This identity is sometimes called Nicomachus's theorem. If you type this in google, you'll recieve numerous pictures.

1. enter image description here

2.enter image description here

3.enter image description here

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An engineering style fourier-approach. You may not consider this to be very intuitively "geometric", but I thought it could be interesting however.

Consider the time-signal $[1,2,...,k]$ (A linear function or "triangle wave"). The LHS is the square of the DC component of the signal in the temporal/spatial domain which is straight forward to calculate. The RHS is the iterated convolution of the fourier transform of [1,2,3,...,k] three times and then DC component of that.

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  • $\begingroup$ Could you please take a look at this $\endgroup$ – Aditya Hase Sep 18 '15 at 1:09

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