<s>[INST] Help with an induction proof Here's the thing I'm trying to prove
$2^k - (2^{k-1} + 2^{k-2} + ... + 2^2 + 2^1) = 2$
It's obviously easy for the k = 1 case, but I'm stuck on the k + 1 case.
 [/INST] Hint: Note that
\begin{align}
2^{k + 1} - \left(2^k + 2^{k - 1} + ... + 2^2 + 2^1\right) &= 2^{k} + 2^k - \left(2^k + 2^{k - 1} + ... + 2^2 + 2^1\right) \\ &= 2^k - \left(2^{k - 1} + ... + 2^1\right)\end{align}
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