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Learn how to prove n! > 2n for n > 3 in natural numbers using mathematical induction. Follow the step-by-step process shown by the Project Maths Development Team in 2011.
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To prove by induction that n! > 2n for n > 3, n N Next (c) Project Maths Development Team 2011
To prove by induction that n! > 2n for n > 3, n N Prove :n! > 2n for n = 4 n! = 4! = 24 24 = 16 24 > 16 True for n = 4 Next (c) Project Maths Development Team 2011
To prove by induction that n! > 2n for n > 3, n N • Assume true for n = k • Therefore k! > 2k • Prove true for n=k+1 Multiply each side by k + 1 (As k>3 hence (k+1) must be positive) (k + 1)k! > (k + 1)2k (k + 1)! >(k + 1)2k (Because (k+1)k!=(k+1)!) • (k + 1)! > 2k + 1(k + 1 > 2 since k > 3) If true for n = k this implies it is true for n = k+1. It is true for n = 4. Hence k! > 2k for n > 3, n N (c) Project Maths Development Team 2011
