Question 12
A sequence of finite sets of positive integers is defined as follows. $S_0 = \{m\}$, where $m \gt 1$. Then given $S_n$ you derive $S_{n + 1}$ by taking $k^2$ and $k + 1$ for each element $k$ of $S_n$. For example, if $S_0 = \{5\}$, then $S_2 = \{7, 26, 36, 625\}$. Show that $S_n$ always has $2^n$ distinct elements.