If the first term of a $G.P.$ is $5$ and the common ratio is $-5$,then which term is $3125$ (in $^{th}$)?

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

Let $a$ and $b$ be two distinct positive real numbers. Let the $11^{\text{th}}$ term of a $GP$,whose first term is $a$ and third term is $b$,be equal to the $p^{\text{th}}$ term of another $GP$,whose first term is $a$ and fifth term is $b$. Then $p$ is equal to:

The sum of the $3^{rd}$ and the $4^{th}$ terms of a $G.P.$ is $60$ and the product of its first three terms is $1000$. If the first term of this $G.P.$ is positive,then its $7^{th}$ term is

If $x$ is added to each of the numbers $3, 9, 21$ so that the resulting numbers are in $G.P.$,then the value of $x$ is:

The $G.M.$ of the numbers $3, 3^2, 3^3, ..., 3^n$ is