I realise that this is supposed to be done with induction. However, I'll just mention a smoother method. You can just consider the equation modulo 3. Then you can sub 0,1,2 into the equation and since and and we are done.
*modulo means considering all numbers as their remainder when divided by 3.
Modulo arithmetic is not taught in the HSC maths courses.
Else I would've used it.
Explanation to the curious ones though:
If you think about it. If you go back to Year 3 and you did long division with integers, if you divided something by 3 your remainder would be either 0, 1 or 2. Hence, every number must be said to be 0, 1, or 2 "modulo 3". The idea is to think only in terms of the remainder as it significantly reduces your cases.
It's one of the instances where proof by exhaustion is swifter than proof by induction.