Steps to Summation closed form

Question

I have $$\sum_{n=1}^k 2^n$$ I got this result from trial and error (validated online), but I want to understand the steps to get there. $$= 2^{k+1}-2$$

Answer 1

This is a geometric series, as each successive term is double the previous term.

You may know a general formula for the sum of a finite geometric series, or you can argue that this particular sum should have value $2^{k+1}-2$ as follows:

$$\begin{align} s=\sum_{i=1}^k 2^i&=2+4+8+\cdots+2^k\\ s&=2(1+2+4+\cdots+2^{k-1})\\ s&=2\left(1+\sum_{i=1}^{k-1}2^i\right)\\ s/2-1+2^k&=\sum_{i=1}^k2^i\\ s/2-1+2^k&=s\\ 2^k-1&=s/2\\ s&=2^{k+1}-2\end{align}$$

This proof generalises to give the general formula.