discrete mathematics - prove using strong mathematical induction $S_{k}=2^{((k+1)/2)}$

I used induction to find the following formula:
$S_{k}=2^{((k+1)/2)}$ for when k is odd. I now need to use induction to prove this formula is right.

I'm not sure how to prove this using strong mathematical induction. Any help is appreciated. I know you are meant to plug (k+2) in for k is that the only difference?

edit - $S_{k}=2S_{k-2}$

Answer

Assume $S_1=2$. Base case is obvious (k=1,2). Let $n\in\mathbb{N}$ and suppose $S_k=2^{(k+1)/2}$ for all $1\leq k\leq n$. Now, it follows that $$S_{n+1}=2S_{n-1}=2(2^{(n)/2})=2^{1+(n)/2}=2^{(n+2)/2}.$$ Hence by PMI, $S_k=2^{(k+1)/2}$ is true for all $k\in\mathbb{N}$. $_\square$

Remarks: Strong induction is just like induction, except you are assuming that $k$ is true for all $1\leq k\leq n$ instead of just $n$.