I have been proving a couple of induction problems.

I am struggling when there are sequences on both sides of the equals.

How do I prove by induction.

$\displaystyle 1^3+2^3+3^3+ ... +n^3={(1+2+3+...+n)}^2$

demonstrating for k=1 is trivial.

Assume for k, and then prove for (k+1).

$\displaystyle 1^3+2^3+3^3+ ... +k^3+{(k+1)}^3={(1+2+3+...+k+(k+1))}^2$

using $\displaystyle \sum\limits_1^n n = \frac{n(n+1)}{2}$

$\displaystyle ={[\frac{k(k+1)}{2} + (k+1)]}^2$

I have tried expanding this, it quickly get's horrible, but also doesn't seem closer to solving the problem.

Thanks

Regards

Craig.