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M42.induction

# M42.induction - MATH 42 Induction The principle of...

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MATH 42 — Induction The principle of mathematical induction refers to a sequence of mathematical statements that we can enumerate with natural numbers. For example, such a sequence of statements is: 1 = 1 2 1 + 3 = 2 2 1 + 3 + 5 = 3 2 1 + 3 + 5 + 7 = 4 2 and so on... which can be summarized as the family of statements 1 + 3 + 5 + · · · + (2 n - 1) = n 2 for every n . The principle of mathematical induction is a means for proving such statements. The steps of an induction proof are as follows. 1. Prove that the first statement is true. 2. Assume the k th statement is true. 3. Prove that the ( k + 1) st statement is true. If the family of statements is a ladder that you’re trying to climb all the way to the n th rung, then Steps 2 and 3 say that if you’re on the k th rung then you can get to the ( k + 1) st rung. The first step says that you can begin your climb! Here is an induction proof that n i =1 (2 i - 1) = n 2 . Step 1: The statement is clearly true when n = 1.

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M42.induction - MATH 42 Induction The principle of...

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