We can derive the general formula for the number of moves by induction. Let us denote $N$ as the number of rings in the problem. $$ $$ For $N=2$, it is straightforward to show that the problem can solved in 3 moves.
For $N=3$, we can see again that 7 moves are required to solve the problem. $$ $$ In general the formula for the number of moves in the Tower of Hanoi problem is $$ n_{\moves} = 2^N - 1. $$ For $N=5$, the number of moves is therefore 63.