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Long time lurker of this forum, but first time poster. Discrete Math is going to be the end of me; I'm just not understanding how to solve problems and write the proofs. Any help would be greatly appreciated. Thanks in advance.

The Problem:

Let nεZ≥1. Show that 1^3+3^3+5^3+.....+(2n-1)^3=n^2(2n^2-1).

Proof:

I'm using induction as this seems like a prime candidate. The claim was obvious, so I found myself at this next step:

(2n-1)^3+(2(n+1)-1)^3=n^2(2n^2-1)+(2(n+1)-1)^3

From here, I just don't see how to algebraically manipulate one side to look like the other. I've been banging my head against the wall on this problem (and others) for hours now.

Help! Thanks again!