The sum of $3$ numbers in geometric progression is $38$ and their product is $1728$. The middle number is

If $\frac{a + bx}{a - bx} = \frac{b + cx}{b - cx} = \frac{c + dx}{c - dx}$ $(x \ne 0)$,then $a, b, c, d$ are in

Let $a_1, a_2, a_3, \ldots$ be a $G.P.$ of increasing positive numbers. Let the sum of its $6^{\text{th}}$ and $8^{\text{th}}$ terms be $2$ and the product of its $3^{\text{rd}}$ and $5^{\text{th}}$ terms be $\frac{1}{9}$. Then $6(a_2 + a_4)(a_4 + a_6)$ is equal to

If the sum of three terms of a $G.P.$ is $19$ and their product is $216$,then the common ratio of the series is

If ${x_1}, {x_2}, {x_3}$ as well as ${y_1}, {y_2}, {y_3}$ are in $G$.$P$. with the same common ratio,then the points $({x_1}, {y_1}), ({x_2}, {y_2})$ and $({x_3}, {y_3})$:

Let $\alpha, \beta, \gamma$ $(\alpha < \beta < \gamma)$ be roots of $ax^3+bx^2+cx+d=0$ and $u, v, w$ $(u < v < w)$ be roots of $ak^3x^3+bk^2x^2+ckx+d=0$. If $\beta^2=\alpha \gamma$,then: