If the sum of the first $p$ terms of an $A.P.$ is equal to the sum of the first $q$ terms,then find the sum of the first $(p+q)$ terms.

The first term of an arithmetic progression is $10$ and the last term is $50$. If the sum of all its terms is $300$,then the number of terms $n = ...$

Let $\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}$ be an $A$.$P$. of four terms such that each term of the $A$.$P$. and its common difference $l$ are integers. If $\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}=48$ and $\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}+l^{4}=361$, then the largest term of the $A$.$P$. is equal to

Let the sequence $a_1, a_2, a_3, \dots, a_{2n}$ form an $A.P.$ Then $a_1^2 - a_2^2 + a_3^2 - a_4^2 + \dots + a_{2n - 1}^2 - a_{2n}^2 = $

Write the first five terms of the sequence whose $n^{th}$ term is $a_{n} = \frac{2n - 3}{6}$.

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