The number which should be added to the numbers $2, 14, 62$ so that the resulting numbers may be in $G.P.$ is

If the roots of the cubic equation $ax^3 + bx^2 + cx + d = 0$ are in $G.P.$,then

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

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

If the product of three terms of a $G.P.$ is $512$. If $8$ is added to the first term and $6$ is added to the second term,the resulting numbers are in $A.P.$. Find the numbers.

Let ${a_n}$ be the ${n^{th}}$ term of a $G$.$P$. of positive numbers. Let $\sum\limits_{n = 1}^{100} {{a_{2n}}} = \alpha$ and $\sum\limits_{n = 1}^{100} {{a_{2n - 1}}} = \beta$,such that $\alpha \ne \beta$,then the common ratio is