If $\frac{1}{2 \cdot 3^{10}}+\frac{1}{2^{2} \cdot 3^{9}}+\ldots+\frac{1}{2^{10} \cdot 3}=\frac{K}{2^{10} \cdot 3^{10}}$,then the remainder when $K$ is divided by $6$ is

The sum of $100$ terms of the series $0.9 + 0.09 + 0.009 + \dots$ will be

For a $G$.$P$.,if $S_n = \frac{4^n - 3^n}{3^n}$,then $t_2 = ........$

The interior angles of an $n$-sided convex polygon are in $G.P.$ The smallest angle is $1^\circ$ and the common ratio is $2$. Then the number of possible values of $n$ is:

The sum of infinite terms of a $G.P.$ is $x$ and on squaring each term of it,the sum becomes $y$. Then the common ratio of this series is:

If $r > 1$,$x = a + \frac{a}{r} + \frac{a}{r^2} + \dots \infty$,$y = b - \frac{b}{r} + \frac{b}{r^2} - \dots \infty$,and $z = c + \frac{c}{r^2} + \frac{c}{r^4} + \dots \infty$,then $\frac{xy}{z} = \dots$