The sum can be found of a infinite $G.P.$ whose common ratio is $r$

The sum of few terms of any ratio series is $728$, if common ratio is $3$ and last term is $486$, then first term of series will be

Find the sum of the products of the corresponding terms of the sequences $2,4,8,16,32$ and $128,32,8,2, \frac{1}{2}$

If $2(y - a)$ is the $H.M.$ between $y - x$ and $y - z$, then $x - a,\;y - a,\;z - a$ are in

Ten trucks, numbered $1$ to $10$ , are carrying packets of sugar. Each packet weights either $999\,g$ or $1000\,g$ and each truck carries only the packets equal weights. The combined weight of $1$ packet selected from the first truck,$2$ packets from the second,$4$ packets from the third, and so on, and $2^9$ packet from the tenth truck is $1022870\,g$. The trucks that have the lighter bags are

Let ${a_n}$ be the ${n^{th}}$ term of the G.P. of positive numbers. Let $\sum\limits_{n = 1}^{100} {{a_{2n}}} = \alpha $ and $\sum\limits_{n = 1}^{100} {{a_{2n - 1}}} = \beta $, such that $\alpha \ne \beta $,then the common ratio is