$\forall n\in\mathbb{N}: n\ge 1 \rightarrow 2^n\le 2^{n+1}-2^{n-1}-1.$
I know the basic part so I won't type it in here, and here is my inductive steps:
$2^{k+1}=2^k\cdot 2 \le 2(2^{k+1}-2^{k-1}-1)$
$2^{k+1} \le 2^{k+2} - 2^k -2$
I know that $2^{k+2}$ and $2^k$ is correct and I can assume $n = k+1$ to get $2^{n+1}-2^{n-1}$
But I don't know how to change $-2 $ to $ -1$ and make it as $2^{n+1}-2^{n-1}-1.$
why do you need to change -2 to -1? you already have:
$2^{k+1} \leq 2^{k+2} - 2^k - 2 \leq 2^{k+2} - 2^k - 1$
Note, dividing by 2 will not work, because you will get the induction assumption... you need to prove $2^{k+1}$ not $2^k$, in fact you assumed the case $2^k$ and used it