If the first term of an infinite geometric series is twice the sum of all its subsequent terms,what is the common ratio?

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

If the $5^{th}$ term of a $G.P.$ is $\frac{1}{3}$ and the $9^{th}$ term is $\frac{16}{243}$,then the $4^{th}$ term will be:

The first term of a $G.P.$ whose second term is $2$ and sum to infinity is $8$,will be

Let $a_1, a_2, a_3, \ldots$ be a $G.P.$ of increasing positive terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$,then $a_6$ is equal to