Simplify $2^{(n-1)} + 2^{(n-2)} + .... + 2 + 1$ Simplify $2^{(n-1)} + 2^{(n-2)} + .... + 2 + 1$
I know the answer is $2^n - 1$, but how to simplify it?

This can be done by induction. The goal is to show that for all $n \ge 0$, $\sum_{i=0}^n 2^i = 2^{n+1}$
First consider the base case $n=0$. This checks out because $2^0 = 1 = 2^1 -1$
Now $\sum_{i=0}^n 2^i = 2^n + \sum_{i=0}^{n-1} 2^i = 2^n +2^n -1 = 2*2^n -1 =2^{n+1} -1$.
