Can I mix direct proof with inductive proof?

Question

Let's say I want to prove with induction that

$3|n$ implies $3|n^2$

Let $n = 3k$. The statement is true for $k=1$ since $3|3$ and $3|9$

We assume the statement is true for $k=z$ so $3|3z$ implies $3|(3z)^2$. From that we know that $(3z)^2=9z^2=3x$ for some integer $x$.

To show that the statement is true for $k=z+1$: $$(3z+3)^2=9z^2+18z+9=3x+18z+9=3(x+6z+3)$$

Is this valid?

Answer 1

Technically, there's nothing wrong with that proof*. There's no reason you can't mix different proof techniques as long as you apply each one correctly - really, induction just demands that you prove the base case and inductive step somehow and there's no reason these might not require somewhat elaborate proofs.

That said, the example proof you give here will surely confuse the reader; if induction isn't necessary, it's generally best not to use it.

(*There is a typo, as $3x+18z+9$ should equal $3(x+6z+3)$, but you've missed a character)