Lesson in an induction problem I'm trying to do this problem but I'm having a basic misunderstanding that just needs some clarification.
Consider the proposition that $P(n) = n^2 + 5n + 1$ is even.
Prove $P(k) \to P(k+1)$ $\forall k \in \mathbb N$.
For which values is this actually true?
What is the moral here?
This problem is meant to tell you a moral problem of induction. I'm aware that $P(n)$ is odd for all integers (I think), so I can't think of where to start on this. This is in regard to induction specifically even if the problem doesn't implicitly state it.

You should be doing the base case before the inductive step. It can be shown that $n=k+1$ follows from $n=k$ using just the inductive step, however that does not imply that the proposition is true.

Here I demonstrate this:
Assume true for $n=k$ where $p\in \mathbb{Z}$:
$$k^2+5k+1=2p$$
For $n=k+1$:
$$(k+1)^2+5(k+1)+1=k^2+2k+1+5k+5+1=(k^2+5k+1)+2k+6$$
Substituting gives:
$$2p+2k+6$$
Hence, this implies that $n=k+1$ follows from $n=k$.

However, note that when we do the base case, $n=1$, we see that this proposition is not true:
$$1^2+5+1=7$$
