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:

Only one disk may be moved at a time.

Each move consists of taking the upper disk from one of the stacks and placing it on

top of another stack or on an empty rod.

No disk may be placed on top of a disk that is smaller than it$.$

Figure $1$: Hanoi Tower Puzzle

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

disks.

Question $1(1$ pt$).$ Explain why for any ninN,$a_{n+1}=2a_n+1.$

Question $2.$ Find an explicit formula $($i$.$e$.,$ not a recursive relation$)$ for $a_n,$ 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.