Directions for the below question:
Let $a_1=p$ and $b_1 =q$ where $p$ and $q$ are positive quantities.
Define:
$a_n pb_{n-1} \: \: \: b_n=qb_{n-1}$ for even $n>1$
and $a_n pa_{n-1} \: \: \: b_n=qa_{n-1}$ for odd $n>1$
If $p=1/3$ and $q= 2/3,$ then what is the smallest odd $n$ such that $a_n + b_n < 0.01$?
- $7$
- $13$
- $11$
- $9$
- $15$