Proof a recursive formula with induction
I'm new here and I could use some help on a homework question.
Given the following recursive definition:
I have to proof, using induction, that
My current proof is as follows:
The formula is correct for the cases n=0 and n=1 (Can be easily verified).
Assume the formula is correct for n, we can fill this in:
However, I think I am missing some important induction steps. I don't see the connection between the base step and the induction step. (Worried)
Any help is appreciated, thanks in advance! (Happy)
Re: Proof a recursive formula with induction
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.