If $s$ is the sum of an infinite $G.P.$ and $a$ is the first term,then the common ratio $r$ is given by:

How many terms of the $G.P.$ $3, \frac{3}{2}, \frac{3}{4}, \ldots$ are needed to give the sum $\frac{3069}{512} ?$

If $x, y, z$ are in geometric progression and $a^x = b^y = c^z$,then . . . . . .

Show that the products of the corresponding terms of the sequences $a, ar, ar^{2}, \dots, ar^{n-1}$ and $A, AR, AR^{2}, \dots, AR^{n-1}$ form a $G.P.$,and find the common ratio.

If $y = x - x^2 + x^3 - x^4 + \dots \infty$,then the value of $x$ will be

If $a, b, c$ and $d$ are in $G.P.$,show that:
$(a^{2}+b^{2}+c^{2})(b^{2}+c^{2}+d^{2})=(ab+bc+cd)^{2}$