<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] As the sum of a geometric sequence, $1+2^1+2^2+\ldots+2^n = \frac{2^{n+1}-1}{2-1}=2^{n+1}-1$.
You can see it by computing $(1+2^1+2^2+\ldots+2^n)(2-1)$ and distribute. All the terms except for $2 \cdot 2^n - 1$ will cancel each other out. 
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