Let $a_1, a_2, a_3, \dots$ be an $A.P.$ such that $\frac{a_1 + a_2 + \dots + a_p}{a_1 + a_2 + \dots + a_q} = \frac{p^3}{q^3}$,where $p \neq q$. Then $\frac{a_6}{a_{21}}$ is equal to:

If the $m^{th}$ terms of the series $63 + 65 + 67 + 69 + \dots$ and $3 + 10 + 17 + 24 + \dots$ are equal,then $m = $

If $a, b, c$ are in $H.P.$,then

The roots of the equation $x^5 - 40x^4 + px^3 + qx^2 + rx + s = 0$ are in $G.P.$ The sum of their reciprocals is $10$. Then the value of $|s|$ is

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball,the second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls used in forming the equilateral triangle,then all these balls can be arranged in a square whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is

If $\sum_{r=1}^{n}r^3 - \sum_{p=1}^{n} \sum_{m=1}^{p} \sum_{r=1}^{m} 1 = 80$,then the possible value of $n$ is: