Using the principle of mathematical induction, prove each of the following for all n ϵ N:
(x2n – 1) - 1 is divisible by (x – y), where x ≠ 1.
To Prove:
Let us prove this question by principle of mathematical induction (PMI) for all natural numbers
Let P(n):
For n = 1
P(n) is true since
which is divisible by x - 1
Assume P(k) is true for some positive integer k , ie,
=
, where m ∈ N …(1)
We will now prove that P(k + 1) is true whenever P( k ) is true
Consider ,
[ Adding and subtracting 1 ]
[ Using 1 ]
, which is factor of ( x - 1 )
Therefore, P (k + 1) is true whenever P(k) is true
By the principle of mathematical induction, P(n) is true for all natural numbers, ie, N.
Hence proved.