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Prove that for all positive integers $n, 1^3+2^3+\ldots+n^3=(1+2+\ldots+n)^2$

Amzoti
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Commander Shepard
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  • Hint: induction – Alexander Aug 06 '13 at 02:23
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    What are your thoughts? You know this is an induction problem. Have you proven anything by mathematical induction before? Are you familiar with common techniques used in these kinds of problems? It would help a lot if you could lay out your thought process. – Alex Wertheim Aug 06 '13 at 02:23

1 Answers1

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Suppose you have proven that $$ 1^3+2^3+\ldots+n^3=(1+2+\ldots+n)^2$$

Note that using the binomial theorem

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

One can use at this point that $1+\cdots+n=\dfrac{n(n+1)}2$ so that the above becomes $$\begin{align}(1+2+\cdots+n+(n+1))^2&=(1+2+\cdots+n)^2+n(n+1)^2+(n+1)^2\\ &=(1+2+\cdots+n)^2+(n+1)(n+1)^2\\ &=(1+2+\cdots+n)^2+(n+1)^3\end{align}$$

and induction kicks in.

Pedro
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