The numbers $(\sqrt{2} + 1), 1, (\sqrt{2} - 1)$ will be in

Find the sum up to $20$ terms in the geometric progression $0.15, 0.015, 0.0015, \dots$

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:

The sum of a few terms of a geometric series is $728$. If the common ratio is $3$ and the last term is $486$,then the first term of the series will be:

Show that the ratio of the sum of the first $n$ terms of a $G.P.$ to the sum of the terms from the $(n+1)^{th}$ to the $(2n)^{th}$ term is $\frac{1}{r^{n}}$.

If the sum of an infinite geometric series is $\frac{4}{5}$ and its $1^{st}$ term is $\frac{3}{4}$,then its common ratio is