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 $S = 2^{0}+ 2^{1}+\ldots+2^{n}.$ Then, $2S = 2^{1}+ 2^{2}+\ldots+2^{n+1}= (S-2^{0})+2^{n+1},$ so $S=2^{n+1}-1$.
