Day: September 19, 2025

61 I know that the sum of powers of $2$ is $2^ {n+1}-1$, and I know the mathematical induction proof. But does anyone know how $2^ {n+1}-1$ comes up in the first place. For example, sum of n numbers is $frac {n (n+1)} {2}$. The idea is that we replicate the set and put it in a rectangle, hence we can do the trick.