Question 163
Suppose $n \geq 3$ is a positive integer. Let $a_{1}\lt a_{2}\lt \cdots\lt a_{n}$ be an increasing sequence of positive real numbers, and let $a_{n+1}=a_{1}$. Prove that
$$
\sum_{k=1}^{n} \frac{a_{k}}{a_{k+1}} \gt \sum_{k=1}^{n} \frac{a_{k+1}}{a_{k}}.
$$
$$
\sum_{k=1}^{n} \frac{a_{k}}{a_{k+1}} \gt \sum_{k=1}^{n} \frac{a_{k+1}}{a_{k}}.
$$