Prove that (4^n)-1 is divisible by 3 for all natural numbers

Prove that (4^n)-1 is divisible by 3 for all natural numbers

We will use induction for this proof, so first we prove the case for the first natural number 1:

(4^1) - 1 = 3 which is divisible by 3.

Now we need to prove the general case for n+1

4^(n+1) - 1

= 4(4^n) - 1

= 4((4^n) - 1) - 1 + 4

= 4((4^n) - 1) + 3

Now both (4^n - 1) and 3 are divisible by 3, thus by induction we can say for all natural numbers 4^n - 1 is divisible by 3.