If $G_1$ and $G_2$ are two geometric means between two numbers and $A$ is the arithmetic mean between them,then find the value of $\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1}$.

Let $a_{1}, a_{2}, a_{3}, \ldots$ be a $G$.$P$. such that $a_{1} < 0$; $a_{1} + a_{2} = 4$ and $a_{3} + a_{4} = 16$. If $\sum_{i=1}^{9} a_{i} = 4 \lambda$,then $\lambda$ is equal to:

The first term of a $G.P.$ is $7$,the last term is $448$ and the sum of all terms is $889$. Then the common ratio is:

If $\alpha, \beta$ are the roots of $x^2-3x+a=0$ and $\gamma, \delta$ are the roots of $x^2-12x+b=0$ and $\alpha, \beta, \gamma, \delta$ in that order form a geometric progression in increasing order with common ratio $r>1$,then $a+b=$

If $\frac{1}{2 \cdot 3^{10}}+\frac{1}{2^{2} \cdot 3^{9}}+\ldots+\frac{1}{2^{10} \cdot 3}=\frac{K}{2^{10} \cdot 3^{10}}$,then the remainder when $K$ is divided by $6$ is

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