How to find a formula for the expression $2^0+2^1+\cdots+2^n$? Is there a way to simply this equation and express it in terms of $n$?
$$2^0+2^1+2^2+2^3+\cdots+2^n$$

Let $S = 2^0+2^1+2^2+2^3+\cdots+2^n$.
In base $2$, it's $S = 1111\dots 111_2$ written with $n+1$ ones.
So $S + 1 = 1000\dots000_2$ (with $n+1$ zeros) $ = 2^{n+1}$.
Hence $S = 2^{n+1}-1$.
