Upper bound of of fib(n+2)

Question

I have a homework problem that is perplexing me because the math is beyond what I have done, although we were told that it was unnecessary to solve this mathematically. Just provide a close upper bound and justify it.

Let $$f(n) = |\{w ∈ \{a, b\}^n : aa \notin w \}|. $$ Provide an asymptotic upper bound on $f$ as $n\to\infty$.

So far: $$ \begin{array}{c|c|c} n & \text{strings} & \text{compared to } 2^n \\\hline 1 & 2 & 2^n - 0 \\ 2 & 3 & 2^n - 1 \\ 3 & 5 & 2^n - 3 \\ 4 & 8 & 2^n - 8 \\ 5 & 13 & 2^n - 18 \\ 6 & 21 & 2^n - 43 \\ \end{array} $$

The math that will give me an exact bound is beyond me. Obviously $O(2^n)$ is an upper bound, though it is not particularly tight.

Any suggestions on what I should try?