You could also do a proof by induction:

Basecase, n=0, 7^0 + 2 = 3, clearly divisible by 3.

Assume 7^n + 2 is divisible by 3, to show that 7^(n+1) + 2 is too.

7^(n+1) + 2 = 7β
7^n + 2 = 6β
7^n + 7^n + 2.

As we already assumed, 7^n + 2 is divisible by 3, and 6β
7^n is clearly divisible by 3 as well. Thus 7^n + 2 is divisible by three for all natural numbers n.