Probably another simple question and possibly not asked too well. I'd like to know why this is true (I'm looking for a simple, possibly combinatorial explanation with a real-world example):

$$\sum\_{i=0}^{n - 1}2^i = 2 ^ n - 1$$

To add more detail, I don't get why $2^{16}$ is the same as $2^{15}+2^{14}+2^{13}...+2^{0} + 1$.