I'm trying to prove $\displaystyle 3|(x^3 - x)$ through induction. (and for 5 in $\displaystyle x^5 - x$ but that's for later)

Done the base case which simplifies to 2

Assume for $\displaystyle x = k$ evaluates to $\displaystyle k(k + 1) = \mathbb{I}$

Inductive prove for $\displaystyle x = k + 1$ evaluates to $\displaystyle (k + 1)(k + 2)$

But I'm guessing you have to add and minus some value before to incorporate the assumption so I'm guessing we do something here:

$\displaystyle \frac{3k^2 + 9k + 6}{3}$

Don't know what exactly, pointers?

Sorry about the mundane question. I'm not wired to think the 2 steps ahead for proofs.