1. Let T (n) be the number of moves in our solution to the n-disc Towers of Hanoi puzzle. Recall that to solve this puzzle, •move the top n −1 discs from the source to the scratch peg, •move disc n from the source to the destination peg, and •move the top n −1 discs from the scratch to the destination peg. The first step takes T (n −1) moves, the second step takes one move, and the third step takes T (n −1) moves again. In other words, T (n) = 2T (n −1) + 1. This applies when n > 0. At zero, we have T (0) = 0, because with zero discs the start and final states are identical; there are no moves to make. Your task is to find a closed-form expression for T (n) (i.e., one that does not use recursion), and prove that it’s correct using induction.

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The computation systems can be defined as the systems that are capable of solving a problem that includes calculations either mathematical or logical, and are able to produce the result as an output. For example, a simple calculator that is given a set of numbers and operators as input and produces the result as an output can be defined as a computational system. The above example is a simple illustration only, the computational systems range from simple devices to complex devices such as the systems that control the SpaceX companies, Starlink satellites which are being quite recently seen in the skies at many places globally.