Start by writing down what it means to say that k is in the truth set: 5^k-2^k is a multiple of 3, or in other words 5^k-2^k=3p, for some integer p.

Now you can write that as 5^k = 2^k +3p. Substitute that expression for 5^k into the right-hand side of the equation 5^(k+1)-2^(k+1)=5(5^k)-2(2^k), and see where that leads you.