$(3)($based on text problem $5\mathrm{.}9)$ Suppose that in a lake a population of trout increases its own numbers by $10$ percent each year. After the births occur each year, $200$ young trout are added $($each year$)$ to build up the population. Let $x_n$ denote the size of the population after $n$ years, starting with a population of $500$ trout.

In the problems below, you can set$-$up and solve a DDE, or just compute directly with the given information.

$($a$)(1$ point$)$ When does $x_{n}2000$ first happen?