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To prove by induction that 2 n ≥ n 2 for n ≥ 4, n N Next (c) Project Maths Development Team 2011

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Prove : 2 n ≥ n 2 is true for n = 4 2 n = 2 4 = 16 n 2 = 4 2 = 16 16 ≥ 16 True for n = 4. To prove by induction that 2 n ≥ n 2 for n ≥ 4, n N Next (c) Project Maths Development Team 2011

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Assume true for n = k 2 k ≥ k 2 Multiply each side by 2 2.2 k ≥ 2k 2 2 k + 1 ≥ k 2 + k 2 (As k ≥ 4 and k N then k 2 >2k + 1.) Therefore 2 k +1 ≥ k 2 + 2k + 1 Hence 2 k +1 ≥ (k + 1) 2 If true for n = k this implies it is true for n = k+1. It is true for n = 4 and so true for n=5 (4+1) and hence by induction (2) n ≥n 2, n ≥ 4, n N. To prove by induction that 2 n ≥ n 2 for n ≥ 4, n N (c) Project Maths Development Team 2011 Prove for n=k+1 Click for proof of this

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Proof by Induction 1.Explanation 1Explanation 1 2.Explanation 2Explanation 2 3.Example DivisionExample Division 4.Example SequencesExample Sequences 5.Example.

Proof by Induction 1.Explanation 1Explanation 1 2.Explanation 2Explanation 2 3.Example DivisionExample Division 4.Example SequencesExample Sequences 5.Example.

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