Find the $25^{th}$ common term of the following $A.P.'s$

$S_1 = 1, 6, 11, .....$

$S_2 = 3, 7, 11, .....$

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

If the sum of the first $n$ terms of the series $\sqrt 3  + \sqrt {75}  + \sqrt {243}  + \sqrt {507}  + ......$ is $435\sqrt 3 $ , then $n$ equals

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

How many terms of the $A.P.$ $-6,-\frac{11}{2},-5, \ldots \ldots$ are needed to give the sum $-25 ?$

The four arithmetic means between $3$ and $23$ are