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Let:

$\displaystyle p<q$ where $\displaystyle p,q\in\mathbb{N}$

and then:

$\displaystyle S=2^p+2^{p+1}+2^{p+2}+\cdots+2^{q}$

$\displaystyle 2S=2^{p+1}+2^{p+2}+2^{p+3}+\cdots+2^{q}+2^{q+1}$

Subtracting the former from the latter, we find:

$\displaystyle S=2^{q+1}-2^p$

Now, let:

$\displaystyle p=\frac{n^2-n}{2},\,q=\frac{n^2+n}{2}-1$

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