Understanding when assumptions are sufficient for induction proofs to be logically correct.

Question

Assuming that we know $P(0)$ and $P(1)$ and $(\forall n \in N) \space P(2n)\Rightarrow P(2n+2)$ is it sufficient to prove $\forall n P(n)$?

Intuitively I think the answer is no, but I can't find a valid argument to "convince myself". I think we would not be able to prove $P(3)$ correct?

Answer 1

The condition $(\forall n \in N )P(2n) \Rightarrow P(2n + 2)$ implies anything only about $P(n)$ for even $n$, so you couldn't possibly prove $P(3), P(5), P(7)$, ..., and it does not matter that you have both $P(0)$ and $P(1)$.