Prove by induction that 6^n-1 is divisible by 5

Given

6ⁿ – 1 is divisible by 5 ∀ n ∈ N

Proof

We will prove the given statement by induction

STEP 1

n = 1

6ⁿ – 1 = 6¹ – 1 = 5

5 is divisible by 5. Therefore, the statement is true for n = 1

STEP 2

Let the given statement be true for n = k

6^k – 1 = 5x

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

6^k+1 – 1 = 6^k*6 – 1

= (5x + 1)*6 – 1

= 30x + 6 – 1

= 30x + 5

= 5*( 6x + 1 )

Thus, 6^k+1 – 1 is divisible by 5.

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.