lec0222 - Mathematical Induction Often times we would like...

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Mathematical Induction Often times, we would like to make a statement about the natural numbers (0,1,2,. ..) such as: for all natural numbers n, the sum of 1+2+3+. ..+n = n(n+1)/2. However, proving such a statement may be difficult if we are not very creative or proficient with algebra. In particular, it would be nice if we could prove a given statement without plugging in every possible value of n (which would clearly take forever. ..) or using terribly clever mathematics. Induction makes this possible. The induction principle is as follows: Let A N. (N is the set of non-negative integers.) where the following two properties hold: 1) 0 A 2) k A k+1 A Then we have that A = N. In layman’s terms, the way we will use this principle to prove statements is the following way. Given an open statement s(n), where n is an arbitrary non- negative integer, we must show these two following things to prove the statement for all non-negative integers n: 1) s(0) 2) s(k) s(k+1) for all non-negative integers k.
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Now, the first step is relatively simple. Since s(0) is a simple statement, you must simply plug in 0 into the open statement and assess the validity of the statement. The second step must be broken down into two steps. Notice
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lec0222 - Mathematical Induction Often times we would like...

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