I have a proof by strong induction question (I won't ask the full question since someone in a previous course got in trouble for it). We basically have to prove, for a recurring sequence, that when $n$ is an odd the result of $S(n)$ is odd and when $n$ is an even number $S(n)$ is even.
I know that for strong induction we have to show multiple basis cases but with the induction step I am not sure in this case whether we have to to do it in two parts one showing that it's odd and one showing that it's even. Could anyone point me to the right direction?