My First Problem!

Algebra Level 3

$\large \frac{a_{1}}{a_{2}} + \frac{a_{2}}{a_{3}} + \frac{a_{3}}{a_{4}}+ \ldots+ \frac{a_{n-1}}{a_{n}} + \frac{a_{n}}{a_{1}}$

Given $a_{1} , a_{2} , a_{3} ,\ldots , a_{n} > 0$ for $n > 2$ . If the expression above has a minimum value of $pn$ for $p$ is a positive integer, then find the value of $p + 1$ .

The answer is 2.

1 solution

Alex Li
Jul 6, 2015

By AM-GM, we have $\frac{\frac{a_1}{a_2}+\frac{a_2}{a_3}+...+\frac{a_n}{a_1}}{n}\ge \sqrt[n]{\frac{a_1}{a_2}\times\frac{a_2}{a_3}\times...\times\frac{a_n}{a_1}}=1$ . Thus, the minimum value of the sum is $n$ .

Equality can be achieved by setting $a_1=a_2=...=a_n$ , so $p=1$ , and the final answer is $\boxed{2}$ .

Moderator note:

Simple standard approach.

nice job bro, thank you.

Muhammad Rizki Fadillah - 5 years, 11 months ago

My First Problem!

The answer is 2.

1 solution

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