In the induction step, you should fix an arbitrary n >= 2 and assume that the claim holds for n - 1 and n - 2. Then you need to prove it for n. In other words, you prove the following statement where P(n) denotes the claim for n: "For all n >= 2, if P(n - 2) and P(n - 1), then P(n)." Since you proved P(0) and P(1), the induction step gives P(2), then from P(1) and P(2) you get P(3) and so on.

The calculations you did are correct.