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

If $y = x - x^2 + x^3 - x^4 + \dots \infty$,then the value of $x$ is equal to:

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 $.........$.

If $1 + r + r^2 + \dots + r^n = (1 + r) (1 + r^2) (1 + r^4) (1 + r^8)$,then what is the value of $n$?

If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of a $G.P.$ are $a$,$b$,and $c$ respectively,then the value of $a^{q - r} b^{r - p} c^{p - q}$ is equal to

If the $4^{th}$,$7^{th}$,and $10^{th}$ terms of a Geometric Progression $(GP)$ are $a, b,$ and $c$ respectively,then: