If in an infinite $G.P.$ first term is equal to the twice of the sum of the remaining terms, then its common ratio is
$1$
$2$
$1/3$
$-1/3$
Show that the products of the corresponding terms of the sequences $a,$ $ar,$ $a r^{2},$ $......a r^{n-1}$ and $A, A R, A R^{2}, \ldots, A R^{n-1}$ form a $G .P.,$ and find the common ratio.
If ${\log _x}a,\;{a^{x/2}}$ and ${\log _b}x$ are in $G.P.$, then $x = $
The number which should be added to the numbers $2, 14, 62$ so that the resulting numbers may be in $G.P.$, is
If the sum of three terms of $G.P.$ is $19$ and product is $216$, then the common ratio of the series is
If $y = x + {x^2} + {x^3} + .......\,\infty ,\,{\rm{then}}\,\,x = $