## Precalculus (6th Edition) Blitzer

$2000$.
From the given information, we can observe that we have an infinite geometric series having initial population $a$: $a,0.9\left( a \right),{{0.9}^{2}}\left( a \right),{{0.9}^{3}}\left( a \right),\cdots$ Here, $a$ is the initial population of flies. Here ${{a}_{1}}=a$ and the common ratio $r=0.9$. We use the formula for the sum of a geometric series with infinite terms ${{S}_{n}}=\frac{{{a}_{1}}}{\left( 1-r \right)}$. Thus \begin{align} & 20000=\frac{a}{\left( 1-0\cdot 9 \right)} \\ & \ \ \ \ \ \ \ a=20000\times 0\cdot 1 \\ & \ \ \ \ \ \ \ a=2000 \\ \end{align}