# 7.RecRelIter - Recurrence Relations In analyzing the Towers...

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Recurrence Relations In analyzing the Towers of Hanoi, we might want to know how many moves it will take. Let T(n) stand for the number of moves it takes to solve the Towers problem for n disks. Then, we have the following formula: T(n) = T(n-1) + 1 + T(n-1) This is because in order to move a tower of n disks, we first move a tower of n-1 disks, which takes T(n-1) moves. Then we move the bottom disk (this is the +1 above), and then we move a tower of n-1 disks again, which takes us T(n-1) moves again. Simplifying, we get: T(n) = 2T(n-1) + 1 Unfortunately, this isn’t terribly helpful to us, because it’s not a formula in terms of n. To get a formula in terms of n, we will use the iteration technique, which simply utilizes the fact that the formula above is true for all positive integers n. We will also use the fact that T(1) = 1, since it takes one move to move a tower of one disk.

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Iterating to Solve the Recurrence T(n) = 2T(n-1) + 1 = 2[2T(n-2) + 1] + 1, because T(n-1) = 2T(n-2) + 1. = 4T(n-2) + 2 + 1
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7.RecRelIter - Recurrence Relations In analyzing the Towers...

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