Sum of the Powers of $2$ Suppose I have a sequence consisting of the first, say, $8$ consecutive powers of $2$ also including $1$: $1,2,4,8,16,32,64,128$. Why is it that for example, $1 + 2 + 4 = 7$ is $1$ less than the next term in the series, $8$? Even if one was to try, for instance, $524,288 + 1,048,576$ ($2^{19}$ and $2^{20}$), the next term ($2^{21}$) will be $2,097,152$, which is one greater than the previous sum. I found this whilst solving the "if you are given a penny each day and it doubles" problem. Can someone please explain this to me? Thank you.

This is a consequence of a well-known formula, but here is an elementary proof for your case, just using distributivity and the fact that $2 - 1 = 1$:
\begin{align}
(1 + 2 + 4 + \dots + 2^n) &= (2-1)(1 + 2 + 4 + \dots + 2^n) \\
&= 2(1 + 2 + 4 + \dots + 2^n) - (1 + 2 + 4 + \dots + 2^n) \\
&= (2 + 4 + 8 + \dots + 2^{n+1}) -  (1 + 2 + 4 + \dots + 2^n) \\
&= 2^{n+1} - 1
\end{align}
