Three numbers are in $G.P.$ such that their sum is $38$ and their product is $1728$. The greatest number among them is

The number of bacteria in a certain culture doubles every hour. If there were $30$ bacteria present in the culture originally,how many bacteria will be present at the end of $2^{\text{nd}}$ hour,$4^{\text{th}}$ hour and $n^{\text{th}}$ hour?

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 $G.M.$ of the numbers $3, 3^2, 3^3, ..., 3^n$ is

$A$ particle starts at the origin and moves $1$ unit horizontally to the right and reaches $P_{1}$, then it moves $\frac{1}{2}$ unit vertically up and reaches $P_{2}$, then it moves $\frac{1}{4}$ unit horizontally to the right and reaches $P_{3}$, then it moves $\frac{1}{8}$ unit vertically down and reaches $P_{4}$, then it moves $\frac{1}{16}$ unit horizontally to the right and reaches $P_{5}$ and so on. Let $P_{n} = (x_{n}, y_{n})$ and $\lim_{n \rightarrow \infty} x_{n} = \alpha$ and $\lim_{n \rightarrow \infty} y_{n} = \beta$. Then, $(\alpha, \beta)$ is

$1 - \frac{2}{3} + \frac{2 \cdot 4}{3 \cdot 6} - \frac{2 \cdot 4 \cdot 6}{3 \cdot 6 \cdot 9} + \ldots \infty =$