given the assumption.

$\displaystyle (n+1)^3 - (n+1) = n^3 + 3n^2 + 3n + 1 - n - 1 = (n^3 - n) + 3(n^2 + n)$

$\displaystyle n^3 - n$ is divisible by 3 by our assumption.

$\displaystyle 3(n^2 + n)$ is divisible by 3.

This means the sum is also divisible by 3.

So we have proved our induction step (n+1)^3 - (n+1) is divisible by 3 (assuming $\displaystyle n^3 - n$ is divisible by 3).

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With the proofs of the base case and inductive step, we have proved by induction:

$\displaystyle n^3 - n$ is divisible by 3 for natural numbers $\displaystyle n \geq 0$