Use Mathematical induction to prove:
$\displaystyle (k^3)=((n^2)(n+1)^2)/4$ when the series starts at k=1 and is for n terms
To prove by mathematical induction: first prove for n = 1 ie. substitute n=1 and show that the formula holds for this value.
Then assume true for n = b: so in this case assume $\displaystyle \sum_{k=1}^{b} = \frac{b^2(b+1)^2}{4}$
Then, using the assumption, prove that the formula holds for n=b+1.
I'm tipping that the Godfather's problem was the third step of the inductive proof. So here's my help to prevent the mathematics getting murdered:
$\displaystyle \frac{b^2(b+1)^2}{4} + (b+1)^3 = \frac{b^2(b+1)^2 + 4(b+1)^3}{4}$
$\displaystyle = \frac{b+1)^2(b^2 + 4[b+1])}{4} = \frac{b+1)^2(b^2 + 4b + 4)}{4} = \frac{(b+1)^2(b+2)^2}{4}$
"Someday, and that day may never come, I'll call upon you to do a service for me. But, until that day, accept this [reply] as a gift ...."