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

2. Prove: 3 is a factor of 4 n -1 for n = 1 4 n -1 = 4 1 -1 = 4 – 1 = 3 True for n = 1 To prove by induction that 3 is a factor of 4 n - 1, n  N Next (c) Project Maths Development Team 2011

3. Assume true for n = k. Therefore 3 is a factor of 4 k -1. Prove true for n = k + 1 For n = k + 1 4 k+1 – 1 = 4(4 k - 1) + 4.1 – 1 (Subtract and add 4.1) = 4(4 k - 1) + 3 (4 k – 1 is divisible by 3 and 3 is divisible by 3.) Hence 4 k+1 – 1 is divisible by 3 when n = k + 1 provided it is divisible by 3 for n = k. It is true for n = 1. Hence by induction 4 n – 1 is divisible by 3 for all n  N. To prove by induction that 3 is a factor of 4 n - 1, n  N (c) Project Maths Development Team 2011