After inserting $n$, $A.M.'s$ between $2$ and $38$, the sum of the resulting progression is $200$. The value of $n$ is

The sum of the numbers between $100$ and $1000$, which is divisible by $9$ will be

The ratio of the sums of first $n$ even numbers and $n$ odd numbers will be

Let $a_1=8, a_2, a_3, \ldots a_n$ be an $A.P.$ If the sum of its first four terms is $50$ and the sum of its last four terms is $170$ , then the product of its middle two terms is

If the sides of a right angled traingle are in $A.P.$, then the sides are proportional to

Let the coefficients of the middle terms in the expansion of $\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4},(1-3 \beta x)^{2}$ and $\left(1-\frac{\beta}{2} x\right)^{6}, \beta>0$, respectively form the first three terms of an $A.P.$ If $d$ is the common difference of this $A.P.$, then $50-\frac{2 d}{\beta^{2}}$ is equal to.