If the $n^{th}$ term of geometric progression $5, - \frac{5}{2},\frac{5}{4}, - \frac{5}{8},...$ is $\frac{5}{{1024}}$, then the value of $n$ is

The third term of a $G.P.$ is the square of first term. If the second term is $8$, then the ${6^{th}}$ term is

If $y = x - {x^2} + {x^3} - {x^4} + ......\infty $, then value of $x$ will be

Suppose that the sides $a,b, c$ of a triangle $A B C$ satisfy $b^2=a c$. Then the set of all possible values of $\frac{\sin A \cot C+\cos A}{\sin B \cot C+\cos B}$ 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 $?$

Find the sum of $n$ terms in the geometric progression $\sqrt{7}, \sqrt{21}, 3 \sqrt{7}, \ldots$