The sum of $n$ terms of two arithmetic progressions are in the ratio $(3n + 8) : (7n + 15)$. Find the ratio of their $12^{\text{th}}$ terms.

For $p, q \in R$,consider the real-valued function $f(x) = (x - p)^2 - q$,where $x \in R$ and $q > 0$. Let $a_1, a_2, a_3, a_4$ be in an arithmetic progression with mean $p$ and a positive common difference $d$. If $|f(a_i)| = 500$ for all $i = 1, 2, 3, 4$,then the absolute difference between the roots of $f(x) = 0$ is:

In an $A.P.$,the sixth term is $a_6 = 2$. If the product $a_1 a_4 a_5$ is the greatest,then the common difference of the $A.P.$ is equal to

The natural numbers are arranged in rows as follows:
$1$
$2, 3$
$4, 5, 6$
$7, 8, 9, 10$
$. . .$
What is the sum of the numbers in the $n^{th}$ row?

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

The sum of the series $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \dots$ to $9$ terms is