Insert two numbers between $3$ and $81$ so that the resulting sequence is a $G.P.$

For $0 < \phi < \frac{\pi}{2}$,if $x = \sum_{n=0}^\infty \cos^{2n}\phi$,$y = \sum_{n=0}^\infty \sin^{2n}\phi$,and $z = \sum_{n=0}^\infty \cos^{2n}\phi \sin^{2n}\phi$,then:

In a $GP$ series consisting of positive terms,each term is equal to the sum of the next two terms. Then,the common ratio of this $GP$ series is

Let $a, ar, ar^2, \ldots$ be an infinite $G.P.$ If $\sum_{n=0}^{\infty} ar^n = 57$ and $\sum_{n=0}^{\infty} a^3 r^{3n} = 9747$,then $a + 18r$ is equal to:

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

If $\frac{a+bx}{a-bx} = \frac{b+cx}{b-cx} = \frac{c+dx}{c-dx}$ and $x \neq 0$,then show that $a, b, c,$ and $d$ are in $G.P.$