If $y = x + {x^2} + {x^3} + .......\,\infty ,\,{\rm{then}}\,\,x = $
$\frac{y}{{1 + y}}$
$\frac{{1 - y}}{y}$
$\frac{y}{{1 - y}}$
None of these
If $x$ is added to each of numbers $3, 9, 21$ so that the resulting numbers may be in $G.P.$, then the value of $x$ will be
Let $x _{1}, x _{2}, x _{3}, \ldots ., x _{20}$ be in geometric progression with $x_{1}=3$ and the common ration $\frac{1}{2}$. A new data is constructed replacing each $x_{i}$ by $\left(x_{i}-i\right)^{2}$. If $\bar{x}$ is the mean of new data, then the greatest integer less than or equal to $\bar{x}$ is $.....$
An $A.P.$, a $G.P.$ and a $H.P.$ have the same first and last terms and the same odd number of terms. The middle terms of the three series are in
The terms of a $G.P.$ are positive. If each term is equal to the sum of two terms that follow it, then the common ratio is
The sum of an infinite geometric series is $3$. A series, which is formed by squares of its terms, have the sum also $3$. First series will be