(c) Project Maths Development Team 2011

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(c) Project Maths Development Team 2011 To prove by induction that 3 is a factor of 4n - 1, n N Next (c) Project Maths Development Team 2011

(c) Project Maths Development Team 2011 To prove by induction that 3 is a factor of 4n - 1, n N Prove: 3 is a factor of 4n -1 for n = 1 4n-1 = 41-1 = 4 – 1 = 3 True for n = 1 Assume true for n = k. Therefore 4k – 1 = 3R.  4k = 3R + 1 . Next (c) Project Maths Development Team 2011

(c) Project Maths Development Team 2011 To prove by induction that 3 is a factor of 4n - 1, n N Prove true for n = k + 1 For n = k + 1 4k+1 – 1 4k 4 – 1 (3R + 1)4 – 1 12R + 4 – 1 12R + 3 3 is a factor of each of these Hence 4k+1 – 1 is divisible by 3 when n = k + 1 by assuming it is true for n = k. It is true for n = 1. Hence by induction 4n – 1 is divisible by 3 for all n N. (c) Project Maths Development Team 2011