Recursive sequence properties

Calculating initial terms: $a_1 = p, b_1 = q$. $a_2 = pq, b_2 = q^2$. $a_3 = p^2q, b_3 = pq^2$. $a_4 = p^2q^2, b_4 = pq^3$. For even $n$: $a_n + b_n = p^{n/2}q^{n/2} + p^{(n-2)/2}q^{n/2} = q(pq)^{\frac{n-1}{2}}(p + q)$.

For odd $n$: $a_n + b_n = (p+q)(pq)^{\frac{n-1}{2}} = 1 \cdot \left(\frac{2}{9}\right)^{\frac{n-1}{2}}$. We want $\left(\frac{2}{9}\right)^{\frac{n-1}{2}} < 0.01$. For $n = 9$, $\frac{n-1}{2} = 4 \implies (2/9)^4 = \frac{16}{6561} \approx 0.0024 < 0.01$. Smallest odd $n$ is 9.