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![proof without words that the sum of the cubes of the first n natural numbers is the square of the sum of the first n natural numbers](http://upload.wikimedia.org/wikipedia/commons/thumb/2/26/Nicomachus_theorem_3D.svg/500px-Nicomachus_theorem_3D.svg.png)
Credit: Cmglee
Nicomachus's theorem states that the sum of the cubes of the first n natural numbers is the square of the sum of the first n natural numbers. This result is generalized by Faulhaber's formula, which gives the sum of pth powers of the first n natural numbers. The special case of Nicomachus's theorem can be proved by mathematical induction, but a more direct proof can be given which is illustrated by a proof without words, pictured here.