Is there a way to simplify the expression $\sum _{k=0}^n 2^k$? That is, is there a way to write it without a $\sum$ or $\prod$ operator?
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What have you tried? Do you notice any patterns when you plug in various values of $n$? – Mark Saving Mar 09 '21 at 19:52
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$2^{n+1}-1$. Sorry I’m just tired. – user56834 Mar 09 '21 at 19:55
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1Does this answer your question? Deriving Sum of a Geometric Progression – Mar 12 '21 at 07:52
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$$1 + 2 + 4 + \cdots 2^n = 11 \cdots 1_\text{binary}=2^{n+1}-1$$
In general, $\sum_{k=0}^n a^n$ is a geometric series.
mjw
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