How do I prove $2^{n+1} + 2n + 1 = 2^{n+2} - 1$ I am attempting to prove using induction:
$\sum_0^n 2i = 2^{n + 1} - 1$
I have gotten to the point where I need to show:
$2^{n+1} + 2n + 1 = 2^{n+2} - 1$
How do I prove this? Or should I be proving the initial question a different way

You missed the power of $2$ in the Left hand side
and the range of $i$ should start from $0$ instead of $1$
We  find  $\displaystyle\sum_0^1 2^i =1+2=3 $ and $ 2^{1 + 1} - 1=4-1=3$
So, $\displaystyle\sum_0^n 2^i = 2^{n + 1} - 1$ is true for $n=1$
Let $\displaystyle\sum_0^n 2^i = 2^{n + 1} - 1$ is true for $n=m$
$$\implies \sum_0^m 2^i = 2^{m + 1} - 1$$
$$\implies \sum_0^{m+1}2^i =\sum_0^m 2^i +2^{m+1}= 2^{m + 1} - 1 +2^{m+1}=2^{m+1}(1+1)-1=2^{m+2}-1$$
