2015/095) If $x^{a}= (x^\frac{b}{2}) (z^\frac{b}{2}) =z^c$ then a,b,c are in?

A) H.P, B) G.P, C) H.P ,D)non of these

we have $x^a = z^c$
so $x =z^\frac{c}{a}\cdots(1)$
further
$x^{a-\frac{b}{2}}=z^{\frac{b}{2}}$
or $x^{2a-b} = z^{b}$
from 1
$z^{(2a-b)*^\frac{c}{a}} = z^{b}$
or
$(2a-b)c= ab$
or
$\dfrac{2}{b} =  \dfrac{1}{a} + \dfrac{1}{c}$
hence they are in HP