Question 9
A sequence of numbers is defined recursively by $a_1 = 1,$ $a_2 = \dfrac{3}{7},$ and
$$
a_n = \dfrac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}
$$
for all $n \ge 3.$ Then $a_{2019}$ can be written as $\dfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. What is $p + q?$
$$
a_n = \dfrac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}
$$
for all $n \ge 3.$ Then $a_{2019}$ can be written as $\dfrac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. What is $p + q?$