If $\frac{1}{\sqrt{b}+\sqrt{c}},\frac{1}{\sqrt{c}+\sqrt{a}},\frac{1}{\sqrt{a}+\sqrt{b}}$ are in A.P., then $9^{ax+1},9^{bx+1},9^{cx+1},$ $x\ne 0$ are in

$\frac{2}{\sqrt{c}+\sqrt{a}}=\frac{1}{\sqrt{b}+\sqrt{c}}+\frac{1}{\sqrt{a}+\sqrt{b}}$$=\frac{2\sqrt{b}+\sqrt{a}+\sqrt{c}}{\left(\sqrt{b}+\sqrt{c}\right)\left(\sqrt{a}+\sqrt{b}\right)}$

Cross multiply 

$\Rightarrow 2\sqrt{ab}+2b+2\sqrt{ac}+2\sqrt{bc}$$=2\sqrt{bc}+2\sqrt{ac}+c+2\sqrt{ab}+a\Rightarrow 2b=a+c$

$\therefore a,b,careinA.P$

$\Rightarrow ax,bx,cxareinA.P$

$\Rightarrow ax+1,bx+1,cx+1areinA.P$

$\Rightarrow 9^{ax+1},9^{bx+1},9^{cx+1}areinG.P$