If the roots of the equation $x^3 - 9x^2 + \alpha x - 15 = 0 $ are in $A.P.$, then $\alpha$ is
$0$
$20$
$21$
$23$
If the sum of three consecutive terms of an $A.P.$ is $51$ and the product of last and first term is $273$, then the numbers are
The number of $5 -$tuples $(a, b, c, d, e)$ of positive integers such that
$I.$ $a, b, c, d, e$ are the measures of angles of a convex pentagon in degrees
$II$. $a \leq b \leq c \leq d \leq e$
$III.$ $a, b, c, d, e$ are in arithmetic progression is
Let $X$ be the set consisting of the first $2018$ terms of the arithmetic progression $1,6,11$,
. . . .and $Y$ be set consisting of the first $2018$ terms of the arithmetic progression $9, 16, 23$,. . . . . Then, the number of elements in the set $X \cup Y$ is. . . .
The arithmetic mean of first $n$ natural number
If $a_m$ denotes the mth term of an $A.P.$ then $a_m$ =