prove by PMI that 2n<n2, is true for all natural numbers, n>=5.

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It must be 2n>n2Pn:  2n>n2 for n5Step 1: Base CaseFor n=525>5232>25 which is true.Hence Pn is true for n=5Step 2: Inductive HypothesisLet Pn be true for n=k2k>nk for some k5Step 3: Inductive Case2k+1=2. 2k>2.k22k+1>2.k2...iNow for n5k-12>2k2+1-2k>2k2>1+2kk2+k2>k2+1+2k2k2>k+12...iiBy i and ii2k+1>k+12Hence Pn is true for n=k+1.Hnece by principal of mathematical induction, Pn is true for all n5.

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