The $6^{th}$ term of a $G.P.$ is $32$ and its $8^{th}$ term is $128$,then the common ratio of the $G.P.$ is

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

If the first term of an infinite geometric series is $1$ and each term is equal to the sum of all the terms that follow it,then what is its fourth term?

$A$ $G.P.$ consists of an even number of terms. If the sum of all the terms is $5$ times the sum of the terms occupying odd places,then the common ratio will be equal to

The sum of the first three terms of a $G.P.$ is $16$ and the sum of the next three terms is $128$. Determine the first term,the common ratio,and the sum to $n$ terms of the $G.P.$

If $a, b$ and $c$ form a geometric progression with common ratio $r$,then the sum of the ordinates of the points of intersection of the line $ax + by + c = 0$ and the curve $x + 2y^2 = 0$ is