The exercise goes like this:
Prove that the statement P(n)
$n^2 + 3*n + 1$ is even
always fails.
My question is if it is sufficient to show that the base case fails for some of the first terms, or is that too trivial and I have to show that it fails when $n$ is odd and when it's even.
Thank you in prior
On
You don't need to show that base case fails for some cases because there is only one base case in this question and that is for $n = 0$ or $n = 1$ or it depends on whatever restriction $n$ has (for instance $n \in \mathbb{N}$). And even if it is too trivial, for the sake of completeness, you should show that the first statement $P(1)$ fails in this case. That's how induction starts.
No it is not enough to prove that it fails just for some small (base) cases. You have to do also IS.
By IH the $P(n)$ is odd, now
$$P(n+1)= (n+1)^2+3(n+1)+1 = (n^2+3n+1)+2n+4 = P(n)+2(n+2)$$