I'm not quite sure what you're doing...? It almost looks like you've assumed the k+1 step, and then did something to it, like working backwards...?

First, you need to show that the equation is true for n = 1.

Once you've done that, assume that the equation is true for some n = k:

$\displaystyle 2(1^3\, +\, 2^3\, +\, ...\, +\, k^3)\, =\, \left[\frac{k(k\, +\, 1)}{2}\right]^2$

You then do the k+1-th step for the left-hand side, and see where that leads:

$\displaystyle 2(1^3\, +\, 2^3\, +\, ...\, +\, k^3\, +\, (k\, +\, 1)^3)$

$\displaystyle 2(1^3\, +\, 2^3\, +\, ...\, \, k^3)\, +\, 2(k\, +\, 1)^3$

Now substitute from the assumption step:

$\displaystyle \left[\frac{k(k\, +\, 1)}{2}\right]^2\, +\, 2(k\, +\, 1)^3$

Convert to a common denominator, combine the terms, and see what you get. It might help to note that you're wanting eventually to end up with:

$\displaystyle \left[\frac{(k\, +\, 1)(k\, +\, 2)}{2}\right]^2$