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Intuitive explanation for the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$

How to prove this without using mathematical induction?

$$1^3+2^3+\cdots+n^3 = (1+2+\cdots+n)^2$$

I know how to prove it by induction, is there a different way?


merged by Eric Naslund Jul 2 '12 at 11:27

This question was merged with Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction because it is an exact duplicate of that question.

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    $\begingroup$ This question is not a duplicate of the question Arjang linked to - generalizations of an identity and alternative proofs of it are different things. $\endgroup$ – Zev Chonoles Jun 27 '12 at 9:28
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    $\begingroup$ @ZevChonoles: But it is a duplicate of both math.stackexchange.com/questions/61482/… and math.stackexchange.com/questions/18548/… I merged those two together , and perhaps will merge this one since there are useful answers here not yet brought up over there. $\endgroup$ – Eric Naslund Jun 27 '12 at 12:48