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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.

principal-ideal-domain
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5 Answers5

25

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

Chappers
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7

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 $x$.

3

page 86 ,85
book :proof without words
author : roger nelsen
you can find here many kind of this proof enter image description here

Khosrotash
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3

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

aGer
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1

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.

mathreadler
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