Monday 22 December 2014

sequences and series - Proving the identity $sum_{k=1}^n {k^3} = big(sum_{k=1}^n kbig)^2$ without induction



I recently proved that




$$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$



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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