Question 18
Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers satisfying $a_{0}=1$ and $a_{n}=a_{\lfloor 7 n / 9\rfloor}+a_{\lfloor n / 9\rfloor}$ for $n=1,2, \ldots$ Prove that there exists a positive integer $k$ with $a_{k}\lt \frac{k}{2001 !}$.
(Here $\lfloor x\rfloor$ denotes the largest integer not greater than $x$.)
(Here $\lfloor x\rfloor$ denotes the largest integer not greater than $x$.)