we can do it my mathematical Induction: the steps as you pkow are
prove that the statement P(n) is true for n=1.
assume that P(k) is true.
finally establish that when P(k) is true it implies P(k+1) is true.
You have established that P(1) is true.
Now we assume P(k) is true that means 7^k + 2 = 3 m for some m in N. That gives 7^k = 3m -2------ 
7^(k+1) + 2 = 7^k * 7 + 2 = 7 ( 3m - 2) + 2 Using (1)
= 21m -14 + 2
I am sure you can take it further