The Tower of Hanoi is a puzzle consisting of three rods and a number of disks of distinct diameters, which can slide onto any rod. The puzzle begins with the disks stacked on one rod in order of decreasing size from the bottom to the top. The objective of the puzzle is to move the entire stack to the last rod, obeying the following rules:

On

Figure $1$: Hanoi Tower Puzzle

Denote by a$,$ the minimal number of moves required to solve a Tower of Hanoi puzzle with n disks.

Question $1(1$ pt$).$ Explain why for any ne N$,20+1.$

Question $2.$ Find an explicit formula $($ie$.,$ not a recursive relation$)$ for a and prove the validity of the formula.

Hint: For Question $2,$ you can use the recursive relation achieved and justified in Question $1.$ for free.