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

2. (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 .
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(c) Project Maths Development Team 2011

3. (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 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