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 $p + 1$ .