If three numbers are in a geometric progression,then their logarithms are in:

If the $p^{\text{th}}$,$q^{\text{th}}$,and $r^{\text{th}}$ terms of a $G.P.$ are $a$,$b$,and $c$ respectively,prove that $a^{q-r} b^{r-p} c^{p-q} = 1$.

The third term of a geometric progression is equal to the square of the first term. If its second term is $8$,then its sixth term will be:

The $4^{\text{th}}$ term of a $GP$ is $500$ and its common ratio is $\frac{1}{m}$,where $m \in N$. 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 $..........$

If the fifth term of a $G.P.$ is $2$,then the product of its first $9$ terms is:

If $5, 5r, 5r^2$ are the lengths of the sides of a triangle,then $r$ cannot be equal to