<s>[INST] 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 $n$?
$$2^0+2^1+2^2+2^3+\cdots+2^n$$
 [/INST] $$\sum\limits_{i=0}^{n}2^i = 1+2+2^2+2^3+\cdots + 2^n = 1 + 2\left(1+2+2^2+\cdots + 2^{n-1}\right)=1+2\sum\limits_{i=0}^{n-1}2^i\text{.}$$
Now subtract $\sum\limits_{i=0}^{n-1}2^i$ from both sides:
$$\sum\limits_{i=0}^{n}2^i-\sum\limits_{i=0}^{n-1}2^i=2^n$$
and 
$$1+2\sum\limits_{i=0}^{n-1}2^i-\sum\limits_{i=0}^{n-1}2^i=1+\sum\limits_{i=0}^{n-1}2^i\text{.}$$
This gives $$2^n-1=\sum\limits_{i=0}^{n-1}2^i$$
or
$$2^{n+1}-1=\sum\limits_{i=0}^{n}2^i\text{.}$$
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