Geometric Series - Powers of 2

The sum $1 + 2 + 4 + 8 + ... + 2^{n-1}$ equals $2^n - 1$. This comes directly from the geometric series formula with $a = 1$ and $r = 2$.

For instance, $1 + 2 + 4 + 8 + 16 = 2^5 - 1 = 31$. This pattern appears when analyzing binary trees (sum of nodes at all levels) or counting subsets (total number of subsets of size $n$).

Notice: the sum of all powers of $2$ up to $2^{n-1}$ is one less than the next power. This relationship is why binary representations work the way they do.