please solve as soon as possible

2. (12 points) (a) The 'Tower of Hanoi' is a mathematical game consisting of three rods and a number of disks of different sizes, which can slide onto any rod. The puzzle starts with the disks in a neat stack in ascending order of size on one rod, the smallest at the top, thus making a conical shape. The aim of the puzzle is to move the entire stack to another rod, obeying the following simple rules: 1. Only one disk can be moved at a time. 2. 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. 3. No larger disk may be placed on top of a smaller disk. With 3 disks, the puzzle can be solved in 7 moves. A game with n = 8 discs is shown below: With n discs the number of moves needed to solve the puzzle is denoted Th. satisfying the following recursively defined sequence: 71 = 1, Tall = 2Th +1, n = 1, 2, ... For further details see https://on. wikipedia. org/wiki/Tower_of_Hanoi# Equivalent_iterative_solution. (i) Using the recursive formula calculate how many moves are needed to solve working!) the puzzle with 2, 3 and 4 discs, i.e., find T, T, and T. (Show your (ii) The formula for the nath term in the sequence, namely Th, is given by Tn = 2" -1, n = 1, 2, .... Verify algebraically that this formula satisfies the recursive sequence given above. (iii) Suppose the time taken to make one move is one second. How long (to the nearest day) would it take to solve a puzzle with 20 discs? (Space for your answers on next page)