1. ## induction proof

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

2. 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.

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 $\displaystyle \sum_{k=1}^{b} k^3 = \frac{b^2(b+1)^2}{4}$ Edit by Mr F: Added the forgotten $\displaystyle k^3$ in the summation.

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:

$\displaystyle \frac{b^2(b+1)^2}{4} + (b+1)^3 = \frac{b^2(b+1)^2 + 4(b+1)^3}{4}$

$\displaystyle = \frac{b+1)^2(b^2 + 4[b+1])}{4} = \frac{b+1)^2(b^2 + 4b + 4)}{4} = \frac{(b+1)^2(b+2)^2}{4}$

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