If $^n{C_4},{\,^n}{C_5},$ and ${\,^n}{C_6},$ are in $A.P.,$ then $n$ can be 

The sum of the first and third term of an arithmetic progression is $12$ and the product of first and second term is $24$, then first term is

Let $AP ( a ; d )$ denote the set of all the terms of an infinite arithmetic progression with first term a and common difference $d >0$. If $\operatorname{AP}(1 ; 3) \cap \operatorname{AP}(2 ; 5) \cap \operatorname{AP}(3 ; 7)=\operatorname{AP}( a ; d )$ then $a + d$ equals. . . . .

If the sum of $n$ terms of an $A.P.$ is $nA + {n^2}B$, where $A,B$ are constants, then its common difference will be

If $a_1, a_2, a_3, …….$ are in $A.P.$ such that $a_1 + a_7 + a_{16} = 40$, then the sum of the first $15$ terms of this $A.P.$ is

If $n$ is the smallest natural number such that $n+2 n+3 n+\ldots+99 n$ is a perfect square, then the number of digits of $n^2$ is