It's a little hard to understand...

What is true for all n? Do you mean that n(n + 1)(n + 2) is divisible by 3 (not n) because one of the factors is a multiple of 3? You are correct, but you have a direct proof rather than a proof by induction because you never use the induction hypothesis. (Coming up with a direct proof is usually more difficult than with a proof by induction.) Indeed, given some n, you have to prove that n³ - n is divisible by 3. Let k = n - 1. Then n³ - n = (k + 1)³ - (k + 1), and then you proceed as before.What I did was expanded (n + 1)³ - (n + 1) and simplified it and I got n(n + 1)(n + 2). From my assumption that it is true for all n, I can see that this is also divisible by n

In a proof by induction, you assume that n³ - n is divisible by 3. Then (n + 1)³ - (n + 1) = n³ - n + 3(n^2 + n), which is divisible by 3 using the assumption.