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.

While you can prove the step that $P(k) \rightarrow P(k+1)$ for all $k \in \mathbb{N}$, it does not follow that "$n^2+5n+1$ is even" for all $n \in \mathbb{N}$.
In other words, the moral is to never forget to prove the base case for induction, for otherwise you might be proving things that are just not true.
Here is a simpler example:
Suppose I want to use induction to prove $P(n): n > n + 1$ for all $n \in \mathbb{N}$
OK, so take arbitrary $k \in \mathbb{N}$, and suppose (inductive hypothesis) that $k > k + 1$. Well, then obviously $k + 1 > (k + 1) + 1$, and so the inductive step is proven. So there!
But wait! Clearly $n > n + 1$ is not true!  What happened? I forgot the base case!
