<s>[INST] The sum of powers of $2$ between $2^0$ and $2^n$ Lately, I was wondering if there exists a closed expression for $2^0+2^1+\cdots+2^n$ for any $n$?
 [/INST] Clue:
If you are searching closed formulas to a series like:
$a+ab+ab^2+ab^3+\ldots+ab^n = S_n$
You can use the follow trick:
$a+ab+ab^2+ab^3+\ldots+ab^n = S_n$
so,
$b(a+ab+ab^2+ab^3+\ldots+ab^n) = bS_n = ab+ab^2+ab^3+ab^4+\ldots+ab^n+ab^{n+1}=$
$a+ab+ab^2+ab^3+\ldots+ab^n -a +ab^{n+1}= bS_n \iff$
$S_n-a+ab^{n+1}=bS_n \iff S_n(b-1)=a(b^{n+1}-b) \iff$
$$S_n=\frac{a(b^{n+1}-b)}{b-1}$$
But, $b=1$ isn't covered by this formula, but if b=1, the series become
$S_n=a+a+a+...+a$ where $a$ appears $n$ times and $S_n=na$.
To your question,
$a=1=2^0$ and $b=2$,
so
$S_n=a+ab+ab^2+ab^3+\ldots+ab^n=2^0+2^1+\ldots+2^n=\frac{1(2^{n+1}-1)}{2-1}=2^{n+1}-1$
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