Show that the ratio of the sum of first $n$ terms of a $G.P.$ to the sum of terms from
$(n+1)^{ th }$ to $(2 n)^{ th }$ term is $\frac{1}{r^{n}}$

$\alpha ,\;\beta $ are the roots of the equation ${x^2} – 3x + a = 0$ and $\gamma ,\;\delta $ are the roots of the equation ${x^2} – 12x + b = 0$. If $\alpha ,\;\beta ,\;\gamma ,\;\delta $ form an increasing $G.P.$, then $(a,\;b) = $

If three geometric means be inserted between $2$ and $32$, then the third geometric mean will be

The product $2^{\frac{1}{4}} \cdot 4^{\frac{1}{16}} \cdot 8^{\frac{1}{48}} \cdot 16^{\frac{1}{128}} \cdot \ldots .$ to $\infty$ is equal to

Find the $12^{\text {th }}$ term of a $G.P.$ whose $8^{\text {th }}$ term is $192$ and the common ratio is $2$