If $a_1, a_2, a_3, \dots$ 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 the roots of the equation $x^{3}+a x^{2}+b x+c=0$ are in $AP$,then $2 a^{3}-9 a b$ is equal to (in $c$)

If the sum of $n$ terms of an $A.P.$ is $S_n = nP + \frac{1}{2}n(n-1)Q$,where $P$ and $Q$ are constants,find the common difference.

The arithmetic mean of the first $n$ even natural numbers is ..........

If the sum of the series $54 + 51 + 48 + \dots$ is $513$,then the number of terms is:

Let $a_1, a_2, a_3, \ldots, a_{100}$ be an arithmetic progression with $a_1=3$ and $S_p=\sum_{i=1}^p a_i, 1 \leq p \leq 100$. For any integer $n$ with $1 \leq n \leq 20$,let $m=5n$. If $\frac{S_m}{S_n}$ does not depend on $n$,then $a_2$ is