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

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

The sum of first three terms of a $G.P.$ is $16$ and the sum of the next three terms is
$128.$ Determine the first term, the common ratio and the sum to $n$ terms of the $G.P.$

The sum of $100$ terms of the series $.9 + .09 + .009.........$ will be

Let ${A_n} = \left( {\frac{3}{4}} \right) - {\left( {\frac{3}{4}} \right)^2} + {\left( {\frac{3}{4}} \right)^3} - ..... + {\left( { - 1} \right)^{n - 1}}{\left( {\frac{3}{4}} \right)^n}$ and $B_n \,= 1 - A_n$ . Then, the least odd natural number $p$ , so that ${B_n} > {A_n}$, for all $n \geq p$ is

If every term of a $G.P.$ with positive terms is the sum of its two previous terms, then the common ratio of the series is