If $S$ is the sum to infinity of a $G.P.$,whose first term is $a$,then the sum of the first $n$ terms is

The sum of an infinite geometric series is $20$,and the sum of the squares of its terms is $100$. Find the common ratio of the geometric series.

$\frac{6}{3^{26}} + \frac{10 \cdot 1}{3^{25}} + \frac{10 \cdot 2}{3^{24}} + \frac{10 \cdot 2^2}{3^{23}} + \ldots + \frac{10 \cdot 2^{24}}{3}$ is equal to

Find the sum to the indicated number of terms in the geometric progression: $x^{3}, x^{5}, x^{7}, \ldots$ for $n$ terms (where $x \neq \pm 1$).

If the $5^{th}$ term of a $G.P.$ is $\frac{1}{3}$ and the $9^{th}$ term is $\frac{16}{243}$,then the $4^{th}$ term will be:

Let $a_1, a_2, a_3, \ldots$ be a $GP$ of increasing positive numbers. If the product of the fourth and sixth terms is $9$ and the sum of the fifth and seventh terms is $24$,then $a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7$ is equal to $.........$.