If the sum of the series $54 + 51 + 48 + .............$ is $513$, then the number of terms are

If $\frac{1}{{p + q}},\;\frac{1}{{r + p}},\;\frac{1}{{q + r}}$ are in $A.P.$, then

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

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

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

Let ${\left( {1 - 2x + 3{x^2}} \right)^{10x}}  = {a_0} + {a_1}x + {a_2}{x^2} + .....+{a_n}{x^n},{a_n} \ne 0$, then the arithmetic mean of $a_0,a_1,a_2,...a_n$ is

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 $3,7,11,15, \ldots, 403$ and $2,5,8,11, \ldots, 404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to.....................