(c) Project Maths Development Team 2011

To prove by induction that 2n ≥ n2 for n ≥ 4, n N

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(c) Project Maths Development Team 2011

Prove : 2n ≥ n2 is true for n = 4 2n = 24 = 16 n2 = 42 = 16 16 ≥ 16

True for n = 4.

To prove by induction that 2n ≥ n2 for n ≥ 4, n N

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(c) Project Maths Development Team 2011

Assume true for n = k 2k ≥ k2

Multiply each side by 2 2.2k ≥ 2k2

2k + 1 ≥ k2 + k2 (As k ≥ 4 and k N then k2>2k + 1.)

Therefore 2k +1 ≥ k2 + 2k + 1 Hence 2k +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 ≥n2, n ≥ 4, n N.

To prove by induction that 2n ≥ n2 for n ≥ 4, n N

Prove for n=k+1

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