This is what I have so far, I'm not sure my reasoning is correct as I am trying to learn how to construct proofs. I would appreciate any feedback on if I took the right steps. If there is an alternative way of going about this problem, what would it be? Thank you in advance.
def. of odd: $i = 2n + 1$
$i^4 = (2n + 1)^4 = (4n^2+4n+1)(4n^2+4n+1)$
From a previous problem, I showed that $4n^2+4n = 8j$. The problem was to show that the square of every odd number is of the form $8k+1.$
$i^4 = (8j+1)(8j+1) = 64j^2+14j+1 = 16j(4j+1)+1$
Here $16j$ is even and $4j+1$ is odd, so multiplying them would yield an even number.
I took a step back and substituted $64j^2+16j=16k$.
Therefore, $i^4 = (4n^2+4n+1) = (8j+1)(8j+1) = 16k+1$.