I don't think you need induction here. Instead, just note that, given three consecutive integers, one must be divisible by 3. Why? Well, let's let the integers be:
n, n+1, n+2
If n is divisible by 3, we're done. Otherwise, n leaves a remainder when divided by 3, either 1 or 2. So n = 3k+1 or n = 3k+2 for some integer k. If n = 3k+1, then n+2 = 3k+3 = 3(k+1) is divisible by 3. If n = 3k+2, then n+1 = 3k+3 = 3(k+1) is divisible by 3.
