Question 135
The two sequences $a_{1}\gt a_{2}\gt \ldots\gt a_{n}$ and $b_{1}\lt b_{2}\lt \ldots\lt b_{n}$ together contain each of the numbers $1,2, \ldots, 2 n$ exactly once. Determine the value of the sum
$$
\left|a_{1}-b_{1}\right|+\left|a_{2}-b_{2}\right|+\ldots+\left|a_{n}-b_{n}\right|
$$
$$
\left|a_{1}-b_{1}\right|+\left|a_{2}-b_{2}\right|+\ldots+\left|a_{n}-b_{n}\right|
$$