$P(0), P(1)$ hold and $P(n) → P(n + 2)$ for $n\geq 1$. For which $n$ is $P(n)$ T?

Question

The question is:

• $P(0)$ hold
• $P(1)$ hold
• $P(n) \rightarrow P(n + 2)$ for $n \geq 1$

For which values of $n$ does $P(n)$ hold?

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My initial answer was that $P(n)$ holds for all odd positive values of $n$, but now I am not so sure.

The reason I thought this was because $n \geq 1$, so I thought $P(0)$ didn't matter.

Is it possible for $P(n)$ to hold for all values of $n \geq 0$, assuming we can start at $P(0)$ or $P(1)$?

Am I completely off base?

Thank you for any help understanding this in advance.

Answer 1

It is possible that $P(n)$ holds for all values $n \ge 0$, simply choose the statement $P(n): 1 = 1$.

However, this is probably not what you actually meant. From your information, you can only infer that $P$ holds for all positive odd integers and for $0$. To see that in general it doesn't hold for any more $n$, consider the example "$P(n)$: $n$ is an odd positive integer or $0$".