If ${\text{a}}^x=b^y=c^z$and $a,b,c$are G.P., then  $x,y,z$ are in 

${\text{a}}^x=b^y=c^z$

$a,b,c$ are G.P.,

$b^2=ac$

$\text{Let }{a}^x=b^y=c^z=k$

$a^x=k\Rightarrow a=k^{\frac{1}{x}}$

$b^y=k\Rightarrow b=k^{\frac{1}{y}}$

$c^x=k\Rightarrow c=k^{\frac{1}{z}}$

${\left(k^{\frac{1}{y}}\right)}^2=k^{\frac{1}{x}}.k^{\frac{1}{z}}$

$\Rightarrow k^{\frac{2}{y}}=k^{\frac{1}{x}+\frac{1}{z}}$

$\Rightarrow \frac{2}{y}=\frac{1}{x}-\frac{1}{z}$

$\Rightarrow \frac{1}{y}+\frac{1}{y}=\frac{1}{x}-\frac{1}{z}$

$\Rightarrow \frac{1}{y}-\frac{1}{x}=\frac{1}{z}-\frac{1}{y}$

$H_3-H_2=H_4-H_3$

$x,y,z$  are in H.P