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 $..........$.

For $p, q \in R$,consider the real-valued function $f(x) = (x - p)^2 - q$,where $x \in R$ and $q > 0$. Let $a_1, a_2, a_3, a_4$ be in an arithmetic progression with mean $p$ and a positive common difference $d$. If $|f(a_i)| = 500$ for all $i = 1, 2, 3, 4$,then the absolute difference between the roots of $f(x) = 0$ is:

Let $x, y > 0$. If $x^{3} y^{2} = 2^{15}$,then the least value of $3x + 2y$ is

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a given $A.P.$ whose common difference is an integer and $S_{n} = a_{1} + a_{2} + \ldots + a_{n}$. If $a_{1} = 1$,$a_{n} = 300$ and $15 \leq n \leq 50$,then the ordered pair $(S_{n-4}, a_{n-4})$ is equal to:

$2+3+5+6+8+9+\ldots$ to $2n$ terms $=$

If the sum of the $10$ terms of an $A.P.$ is $4$ times the sum of its $5$ terms,then the ratio of the first term to the common difference is: