The sum of all integers between $1$ and $100$ (both inclusive) which are divisible by $5$ or $13$ is

If the roots of the equation $x^3 - 9x^2 + \alpha x - 15 = 0$ are in $A.P.$,then $\alpha$ is:

If the sum of the series $20 + 19 \frac{3}{5} + 19 \frac{1}{5} + 18 \frac{4}{5} + \ldots$ up to $n^{\text{th}}$ term is $488$ and the $n^{\text{th}}$ term is negative,then:

Let the terms of an arithmetic progression be $a_1, a_2, a_3, \dots$. If $\frac{a_1 + a_2 + \dots + a_p}{a_1 + a_2 + \dots + a_q} = \frac{p^2}{q^2}$,where $p \neq q$,then $\frac{a_6}{a_{21}} = \dots$

If the sum of the first $n$ terms of an arithmetic progression is $cn^2$,what is the sum of the squares of these $n$ terms?

If $\log_{3} 2, \log_{3} (2^{x} - 5)$ and $\log_{3} (2^{x} - \frac{7}{2})$ are in an Arithmetic Progression $(AP)$,then $x = \dots$