# If $a_{1},a_{2},a_{3},....a_{n}$ are positive real numbers,then

Q) If $$a_{1},a_{2},a_{3},....a_{n}$$ are positive real numbers,then

$$\frac{a_{1}}{a_{2}} + \frac{a_{2}}{a_{3}}+......+\frac{a_{n-1}}{a_{n}}+\frac{a_{n}}{a_{1}}$$

is always

(A) $$\geq n \;$$ (B) $$\leq n\;$$ (C) $$\leq n^{1/n}\;$$ (D) none of the above

I have taken some small values of $$n$$ like $$n=2,3$$ and sample inputs as $$(a_1,a_2,a_3)=(2,3,5)$$,$$(a_1,a_2,a_3)=(1/2,1/3,1/5)$$ , $$(a_1,a_2,a_3)=(0.1,0.2,0.3)$$, $$(a_1,a_2)=(2,3)$$ for which I am getting answer as $$(A)$$ but I am not sure that it is always true.If it is not true always then answer will be $$(D).$$ I don't know how to prove it formally. Please help.

• Use AM-GM. First will be right answer. Mar 17, 2019 at 7:09
• Thank you @Love Invariants, got it..Now I can continue. Mar 17, 2019 at 7:13
• @LoveInvariants Why are you answering in a comment? Mar 17, 2019 at 7:15
• @Arthur-Because a hint would satisfy. OP should do the rest of the work. It helps askers very much. Mar 17, 2019 at 7:16
• @LoveInvariants Hints are still answers more than they are clarification requests. So they belong in answer posts, not in comments. Also, you have a direct answer to the question by saying which alternative was right, so calling it just a hint isn't quite right. Mar 17, 2019 at 7:18

This may be solved using the rearrangement inequality. We have the two sequences $$a_1, a_2, a_3,\ldots,a_n\\ \frac1{a_1}, \frac1{a_2},\frac1{a_3}, \ldots,\frac1{a_n}$$ If we multiply each $$a_i$$ with $$\frac1{a_i}$$, then the sum is $$n$$. By the rearrangement inequality, this is the smallest possible value we can get, so rearranging must give us a result $$\geq n$$.