2 Mathematical induction is a form of mathematical proof.Just because a rule, pattern, or formula seems towork for several values of n, you cannot simplydecide that it is valid for all values of n withoutgoing through a legitimate proof.The Principle of Mathematical InductionLet Pn be a statement involving the positiveinteger n. IfP1 is true, andthe truth of Pk implies the truth of Pk+1 , forevery positive integer k,then Pn must be true for all integers n.
3 Ex. Use mathematical induction to prove the following formula.Sn = (2n-1) = n2First, we must show that the formula works for n = 1.For n = 1S1 = 1 = 12The second part of mathematical induction has twosteps. The first step is to assume that the formula isvalid for some integer k. The second step is to use thisassumption to prove that the formula is valid for thenext integer, k + 1.Assume Sk = (2k-1) = k2is true, show that Sk+1 = (k + 1)2 is true.
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