In a $GP$ series consisting of positive terms, each term is equal to the sum of the next two terms. Then, the common ratio of this $GP$ series is

In a $G.P.$,the $3^{rd}$ term is $24$ and the $6^{th}$ term is $192$. Find the $10^{th}$ term.

Let $f(x)$ be a function such that $f(x+y)=f(x) \cdot f(y)$ for all $x, y \in \mathbb{N}$. If $f(1)=3$ and $\sum_{k=1}^{n} f(k)=3279$,then the value of $n$ is:

If $S$ is the sum,$P$ is the product,and $R$ is the sum of the reciprocals of $n$ terms of a geometric progression,then $P^2 = \dots$

For the functions $f(\theta) = \alpha \tan^2 \theta + \beta \cot^2 \theta$ and $g(\theta) = \alpha \sin^2 \theta + \beta \cos^2 \theta$, where $\alpha > \beta > 0$, let $\min_{0 < \theta < \pi/2} f(\theta) = \max_{0 < \theta < \pi} g(\theta)$. If the first term of a $G$.$P$. is $(\frac{\alpha}{2\beta})$, its common ratio is $(\frac{2\beta}{\alpha})$ and the sum of its first $10$ terms is $\frac{m}{n}$, where $\gcd(m,n)=1$, then $m+n$ is equal to . . . . . . .

If $a, b$ and $c$ form a geometric progression with common ratio $r$,then the sum of the ordinates of the points of intersection of the line $ax + by + c = 0$ and the curve $x + 2y^2 = 0$ is