#### Answer

$x = 10 \sqrt {10}$

#### Work Step by Step

The geometric mean of two numbers can be found using the following proportion:
$\frac{a}{x} = \frac{x}{b}$, where $a$ and $b$ are positive numbers and $x$ is the geometric mean.
Let's plug in our numbers:
$\frac{1}{x} = \frac{x}{1000}$
Use the cross products property to get rid of the fractions:
$x^2 = 1000$
Rewrite $1000$ as the product of a perfect square and another factor:
$x^2 = 100 • 10$
Take the positive square root of each factor to solve for $x$:
$x = 10 \sqrt {10}$