The geometric mean of the sequence $1, 2, 2^2, ..., 2^n$ is:

For the functions $f(\theta) = \alpha \tan^2 \theta + \beta \cot^2 \theta$ and $g(\theta) = \alpha \sin^2 \theta + \beta \cos^2 \theta$, where $\alpha > \beta > 0$, let $\min_{0 < \theta < \pi/2} f(\theta) = \max_{0 < \theta < \pi} g(\theta)$. If the first term of a $G$.$P$. is $(\frac{\alpha}{2\beta})$, its common ratio is $(\frac{2\beta}{\alpha})$ and the sum of its first $10$ terms is $\frac{m}{n}$, where $\gcd(m,n)=1$, then $m+n$ is equal to . . . . . . .

The number $111...1$ ($91$ times) is a

If $x_1, x_2, x_3$ as well as $y_1, y_2, y_3$ are in geometric progression with the same common ratio,then the points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ are

The quotient when $1+x^2+x^4+x^6+\ldots+x^{34}$ is divided by $1+x+x^2+x^3+\ldots+x^{17}$ is

The terms of a $G.P.$ are positive. If each term is equal to the sum of the two terms that follow it,then the common ratio is: