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4 - A proof by mathematical induction(POMI To prove “P(n...

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A proof by mathematical induction (POMI): To prove “ P ( n ) is true for all n 1”: Base Case : Check P (1). Induction Hypothesis : Assume that P ( k ) is true for some k 1. Induction Conclusion : Prove that P ( k + 1) is true using Ind. Hyp. 1
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Example. 2 n > 4 n for all n 5. Base Case : n = 5 : 2 5 = 32 > 20 = 4 · 5. Induction Hypothesis : Assume that 2 k > 4 k for some k 5. Induction Conclusion : We prove 2 k +1 > 4( k + 1) 2 k +1 = 2 k + 2 k (since k 5 2) 2 k + 2 2 > (by Ind.Hyp. ) 4 k + 4 = 4( k + 1) . 2
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A proof by strong induction (POSI): To prove “ P ( n ) is true for all n 1”: Base Case : Check P (1). Induction Hypothesis : Assume P (1) , P (2) , . . . , P ( k ) for some k 1. Induction Conclusion : Prove that P ( k + 1) is true using Ind. Hyp. 3
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To prove “ P ( n ) is true for all n 1”: Check P (1) , P (2) , . . . P ( m - 1) , and prove by induction P ( n ) is true for all n m Base Case : Check P ( m ). Induction Hypothesis : Assume P (1) , P (2) , . . . , P ( k ) for some k m . Induction Conclusion : Prove that P ( k + 1) is true using Ind. Hyp.
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