If ${\log _x}a,\;{a^{x/2}}$ and ${\log _b}x$ are in $G.P.$, then $x = $

If $y = x + {x^2} + {x^3} + .......\,\infty ,\,{\rm{then}}\,\,x = $

The number of natural number $n$ in the interval $[1005, 2010]$ for which the polynomial. $1+x+x^2+x^3+\ldots+x^{n-1}$ divides the polynomial $1+x^2+x^4+x^6+\ldots+x^{2010}$ is

If the sum of first 6 term is $9$ times to the sum of first $3$ terms of the same $G.P.$, then the common ratio of the series will be

The product of three geometric means between $4$ and $\frac{1}{4}$ will be

The $4^{\text {tht }}$ term of $GP$ is $500$ and its common ratio is $\frac{1}{m}, m \in N$. Let $S_n$ denote the sum of the first $n$ terms of this GP. If $S_6 > S_5+1$ and $S_7 < S_6+\frac{1}{2}$, then the number of possible values of $m$ is $..........$