If the roots of the equation $x^3-13x^2+Kx-27=0$ are in geometric progression,then $K=$

What is the product of $n$ geometric means between $a$ and $b$?

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

If there are $5$ geometric means between $486$ and $\frac{2}{3}$,what is the $4^{th}$ geometric mean?

Let $A_{1}, A_{2}, A_{3}, \ldots$ be squares such that for each $n \geq 1,$ the length of the side of $A_{n}$ equals the length of the diagonal of $A_{n+1}$. If the side length of $A_{1}$ is $12 \text{ cm}$,then the smallest value of $n$ for which the area of $A_{n}$ is less than $1 \text{ cm}^2$ is:

If $x = \frac{4}{3} - \frac{4x}{9} + \frac{4x^2}{27} - \dots \infty$,then $x$ is equal to