Question 35
Let $a_1, a_2, ..., a_n, b_1, b_2, ..., b_n$ be the numbers $1, 2, ..., 2n$ in some order. Suppose that $a_1 \lt a_2 \lt ... \lt a_n$ and that $b_1 \gt b_2 \gt ... \gt b_n$. Prove that
$$
\sum_{i=1}^{n} |a_i - b_i| = n^2.
$$
$$
\sum_{i=1}^{n} |a_i - b_i| = n^2.
$$