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 $.....$
$143$
$144$
$145$
$142$
If the sum of an infinite $G.P.$ and the sum of square of its terms is $3$, then the common ratio of the first series is
The geometric series $a + ar + ar^2 + ar^3 +..... \infty$ has sum $7$ and the terms involving odd powers of $r$ has sum $'3'$, then the value of $(a^2 -r^2)$ is -
Find the sum of $n$ terms in the geometric progression $\sqrt{7}, \sqrt{21}, 3 \sqrt{7}, \ldots$
The sum of first three terms of a $G.P.$ is $\frac{39}{10}$ and their product is $1 .$ Find the common ratio and the terms.
The sum to infinity of the progression $9 - 3 + 1 - \frac{1}{3} + .....$ is