Monday, 22 December 2014

sequences and series - Proving the identity sumnk=1k3=big(sumnk=1kbig)2 without induction



I recently proved that




nk=1k3=(nk=1k)2



using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any other proofs.


Answer



Stare at the following image, taken from this MO answer, long enough:



Proof that the sum of the cubes is the square of the sum


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