Prove 4^n-1 is divisible by 3

Given

4n – 1 is divisible by 3 ∀ n ∈ N

Proof

We will prove the given statement by induction

STEP 1

n = 1

4n – 1 = 41 – 1 = 3

3 is divisible by 3. Therefore, the statement is true for n = 1

STEP 2

Let the given statement be true for n = k

4k – 1 = 3x

Now, we need to prove that if the statement is true for n = k then it is also true for n = k + 1

4k+1 – 1 = 4k*4 – 1

= (3x + 1)*4 – 1

= 12x + 4 – 1

= 12x + 3

= 3*( 4x + 1 )

Thus, 4k+1 – 1 is divisible by 3.

Therefore, we can say that if the given statement is true for n = k, then it is also true for n = k + 1. Hence, by the principle of mathematical induction, the given statement is true ∀ n ∈ N.

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