Cube related to squares!

Posted by Nalin Pithwa

Problem:

Show that the cube of a positive integer can always be written as the difference of two squares!

Proof:

We only need to observe that

$n^{3} = (1^{3}+2^{3}+3^{3}+ \ldots + n^{3}) - (1^{3}+2^{3}+3^{3}+ \ldots + (n-1)^{3})$

which, in turn, equals $(\frac{n(n+1)}{2})^{2}-(\frac{(n-1)(n)}{2})^{2}$ .

Hooray ! 🙂 That’s all !

Nalin Pithwa

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