If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an Arithmetic Progression are $a$,$b$,and $c$ respectively,then $[a(q - r) + b(r - p) + c(p - q)] = ?$

Let $f(x)$ be a quadratic polynomial. If $f(1) = f(-1)$ and $a, b, c$ are in arithmetic progression,then $f'(a), f'(b), f'(c)$ are in..... progression.

The sums of $n$ terms of two arithmetic series are in the ratio $(2n + 3) : (6n + 5)$. Then the ratio of their $13^{th}$ terms is:

If the sides of a right-angled triangle are in an arithmetic progression,then they are in the ratio of.........

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.

Let $A = \{1, a_{1}, a_{2}, \ldots, a_{18}, 77\}$ be a set of integers with $1 < a_{1} < a_{2} < \ldots < a_{18} < 77$. Let the set $A + A = \{x + y : x, y \in A\}$ contain exactly $39$ elements. Then,the value of $a_{1} + a_{2} + \ldots + a_{18}$ is equal to: