To Prove
4n – 3n – 1 is divisible by 9 ∀ n ∈ N
Proof
We will prove the given statement using mathematical induction.
Step 1
First, we will check if the given statement is true for n = 1.
4n – 3n – 1 = 41 – 3(1) – 1 = 4 – 3 – 1 = 0
The statement is true for n = 1 since 0 is divisible by 9.
Step 2
In this step, we will assume that the statement is true for n = k. This implies that 4k – 3k – 1 is divisible by 9. We can say
4k – 3k – 1 = 9x, ∀ x ∈ N ……(1)
Step 3
In this step, we will check if the statement is true for n = k + 1 given that it is true for n = k.
Substituting equation (1) in (2)
9(4x+k) is divisible by 9. This implies that the given statement is true for n = k + 1 given that the statement is true for n = k. Therefore, by the principle of mathematical induction, the given statement “4n – 3n – 1 is divisible by 9″ is true for all n ∈ N.