If $a$,$b$,and $c$ are positive real numbers,then $a/b + b/c + c/a$ is greater than or equal to what value?

The number of terms common to the two $A$.$P$.'s $3, 7, 11, \ldots, 407$ and $2, 9, 16, \ldots, 709$ is

The number of terms of the $A.P. 3, 7, 11, 15, ...$ to be taken so that the sum is $406$ is

The $20^{th}$ term from the end of the arithmetic progression $3 + 7 + 11 + \dots + 407$ is ...... .

Let $a_1, a_2, \ldots, a_n$ be in an $A$.$P$. If $a_5 = 2a_3$ and $a_{11} = 18$,then $12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}} + \frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}} + \ldots + \frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$ is equal to $..........$.

If $2x, x + 8$ and $3x + 1$ are in an arithmetic progression,then $x = \dots$