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.
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1I presume the next-to-last term is $\dfrac{a_{n-1}}{a_n}$? – John Wayland Bales Jun 30 '17 at 19:48
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yup @john wayland – Akash Banerjee Jul 01 '17 at 05:03
2 Answers
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You seek $n$ times the arithmetic mean of positive reals of known geometric mean, so by the AM-GM inequality we obtain the minimum when each fraction is equal to that geometric mean. I'll leave the rest as an exercise.
J.G.
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By the inequality of arithmetic and geometric means we have that $$\frac{a_{1}}{a_{2}} + \frac{a_{2}}{a_{3}}+\dots +\frac{a_{n}}{a_{1}} \geq n \, \left( \frac{a_{1}\cdot a_{2}\cdot \dots \cdot a_{n-1} \cdot a_{n}}{a_{2}\cdot a_{3} \dots \cdot a_{n} \cdot a_{1}}\right)^{1/n} = n $$
TheOscillator
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you left no work for the OP -- doing their homework for them is not exactly the point of the site... – gt6989b Jun 30 '17 at 19:49
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I don't -- but the chances are quite high. Neither do you know that it wasn't -- so why take a chance at spoiling the learning process? In such cases, I would rather give a hint, or even a detailed hint -- but solving the whole thing without the OP showing any effort on the problem whatsoever is beyond my comfort level. – gt6989b Jun 30 '17 at 19:58