If the $p^{th}$ term of an arithmetic progression is $q$ and its $q^{th}$ term is $p$,then what is its $(p + q)^{th}$ term?

If $a_1, a_2, \dots, a_n$ are positive real numbers such that their product is a constant $c$,then what is the minimum value of $a_1 + a_2 + \dots + a_{n-1} + 2a_n$?

If the roots of the equation $x^3+ax^2+bx+c=0$ are in arithmetic progression,then

The first term of an $A.P.$ of consecutive integers is ${p^2} + 1$. The sum of $(2p + 1)$ terms of this series can be expressed as:

$a_1, a_2, a_3, \dots, a_{100}$ are in an arithmetic progression,where $a_1 = 3$ and $S_p = \sum_{i=1}^p a_i, 1 \le p \le 100$. For any integer $n$,let $m = 5n$. If $S_m/S_n$ is independent of $n$,then $a_2 = \dots$

Let $a_1, a_2, a_3, \ldots, a_n$ be positive real numbers. Then the minimum value of $\frac{a_1}{a_2}+\frac{a_2}{a_3}+\ldots+\frac{a_n}{a_1}$ is