The arithmetic mean of two numbers $b$ and $c$ is $a$,and $g_1$ and $g_2$ are two geometric means between them. If $g_1^3 + g_2^3 = kabc$,then $k = \dots$

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 $\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}$ and $x \neq 0$,then $a, b, c, d$ are in

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

Three numbers are in $G.P.$ such that their sum is $38$ and their product is $1728$. The greatest number among them is

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