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M115A-AdditionalInductionExample

M115A-AdditionalInductionExample - MATH 115A Additional...

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MATH 115A - Additional Example of the Principle of Mathematical Induction Paul Skoufranis September 28, 2011 In this document we will present an additional example to demonstrate the Principle of Mathematical Induction. Example) You are given three pegs. On one of the pegs is a tower made up of n rings placed on top of one another so that, as you move down the tower, each successive ring has a larger diameter than the previous ring. The object of the puzzle is to reconstruct the tower on one of the other pegs by mov- ing one ring at a time from one peg to another in such a manner that you never have a larger ring above any smaller ring on any of the 3 pegs. Prove that, for any n N , the puzzle can be completed in 2 n - 1 move. Proof : We will apply the Principle of Mathematical Induction to the statements P n where P n is statement that the puzzle with n rings can be solved in 2 n - 1 steps. Base Case: n = 1 For n = 1, there is only one ring. We can clearly move this ring to another peg in 1 = 2 1 - 1 steps. Hence we have proven the base case.
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