If $s$ is the sum of an infinite $G.P.$ and $a$ is the first term,then the common ratio $r$ is given by:

If $a_1, a_2, \dots, a_{50}$ are in a geometric progression,then $\frac{a_1 - a_3 + a_5 - \dots + a_{49}}{a_2 - a_4 + a_6 - \dots + a_{50}} = \dots$

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

The arithmetic mean of two numbers $b$ and $c$ is $a$,and $g_1$ and $g_2$ are two geometric means between them. If $g_1^3 + g_2^3 = kabc$,then $k = \dots$

If $r > 1$,$x = a + \frac{a}{r} + \frac{a}{r^2} + \dots \infty$,$y = b - \frac{b}{r} + \frac{b}{r^2} - \dots \infty$,and $z = c + \frac{c}{r^2} + \frac{c}{r^4} + \dots \infty$,then $\frac{xy}{z} = \dots$

Find the $9^{th}$ term of the geometric progression $8 + 12 + 18 + 27 + \dots$