<s>[INST] What is the minimum value of $\frac{a_1}{a_2} + \frac{a_2}{a_3} +\cdots + \frac{a_n}{a_1} $ Let $a_1,a_2,\cdots,a_n$ are positive real numbers.
Question: what is the minimum value of
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} +\cdots + \frac{a_n}{a_1} $$
Thought: I have no clue how to proceed. Tried some standard inequalities but in vain.
 [/INST] By the inequality of arithmetic and geometric means we have that  $$\frac{a_{1}}{a_{2}} + \frac{a_{2}}{a_{3}}+\dots +\frac{a_{n}}{a_{1}} \geq n \, \left( \frac{a_{1}\cdot a_{2}\cdot \dots \cdot a_{n-1} \cdot a_{n}}{a_{2}\cdot a_{3} \dots \cdot a_{n} \cdot a_{1}}\right)^{1/n} = n $$ 
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