The sum of the first n cubes

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archive posted 4 weeks ago

Prove that for every positive integer $n$

1^{3} + 2^{3} + \dots + n^{3} = (1 + 2 + \dots + n)^{2} .

Posed by Nicomachus of Gerasa, around 100 AD

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algebra induction

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The identity was known to Nicomachus of Gerasa in the first or second century AD. Induction proves it in two lines, and there is a picture proof in which squares of side $1, 2, \dots, n$ fill a square of side $n (n + 1) /2$ .

References

Squared triangular number

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