Question

The value of $1000\prod _{n=1}^9{\left(\frac{n}{n+1}\right)}^2$ (where $\prod$ is product function) is

Answer 1

$\prod _{n=1}^9{\left(\frac{n}{n+1}\right)}^2$
$={\left(\frac{1}{2}\right)}^2\times {\left(\frac{2}{3}\right)}^2\times {\left(\frac{3}{4}\right)}^2\times ...\times {\left(\frac{9}{10}\right)}^2$
$=\frac{1^2}{{10}^2}=\frac{1}{100}$

$\Rightarrow 1000\prod _{n=1}^9{\left(\frac{n}{n+1}\right)}^2=1000\times \frac{1}{100}=10$