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

If the sum of the first $n$ even natural numbers is $k$ times the sum of the first $n$ odd natural numbers,then $k = ........$

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10} = 390$ and the ratio of the tenth and the fifth terms is $15:7$,then $S_{15} - S_5$ is equal to:

If the sum of two extreme numbers of an $A.P.$ with four terms is $8$ and the product of the remaining two middle terms is $15$,then the greatest number of the series will be:

In an arithmetic progression,the sum of the first and third terms is $12$,and the product of the first and second terms is $24$. Find the first term.

If three positive real numbers $a, b, c$ are in $A$.$P$. and $abc = 4$, then the minimum possible value of $b$ is: