Question 1025317
Assuming *[tex \large a_1, a_2, \ldots, a_n] are positive (otherwise the inequality might not hold), then by the AM-GM inequality we have


*[tex \large \frac{ \frac{a_1}{a_2} + \frac{a_2}{a_3} + \ldots + \frac{a_n}{a_1}}{n} \ge \sqrt[n]{\frac{a_1}{a_2} \cdot \frac{a_2}{a_3} \cdot \ldots \cdot \frac{a_n}{a_1}}]


Right-hand side equals 1.


*[tex \large\frac{ \frac{a_1}{a_2} + \frac{a_2}{a_3} + \ldots + \frac{a_n}{a_1}}{n} \ge 1]


*[tex \large \frac{a_1}{a_2} + \frac{a_2}{a_3} + \ldots + \frac{a_n}{a_1} \ge n]


as desired.