For a $G.P.$,if $(m+n)^{\text{th}}$ term is $p$ and $(m-n)^{\text{th}}$ term is $q$,then the $m^{\text{th}}$ term is $.........$

If each term of a geometric progression $a_1, a_2, a_3, \ldots$ with $a_1 = \frac{1}{8}$ and $a_2 \neq a_1$ is the arithmetic mean of the next two terms and $S_n = a_1 + a_2 + \ldots + a_n$,then $S_{20} - S_{18}$ is equal to

If $1 + \sin x + \sin^2 x + \dots \infty = 4 + 2\sqrt{3}$ for $0 < x < \pi$,then:

Let $a_1, \frac{a_2}{2}, \frac{a_3}{2^2}, \ldots, \frac{a_{10}}{2^9}$ be a $G$.$P$. with common ratio $r = \frac{1}{\sqrt{2}}$. If $a_1 + a_2 + \ldots + a_{10} = 62$, then $a_1$ is equal to:

Three numbers are in $G.P.$ such that their sum is $38$ and their product is $1728$. The greatest number among them is

If $5, 5r, 5r^2$ are the lengths of the sides of a triangle,then $r$ cannot be equal to