$a, b, c$ are in AP
$\Rightarrow \sin \mathrm{A}, \sin \mathrm{B}, \sin \mathrm{C}$ are in $\mathrm{AP}$
$\Rightarrow \quad 2 \sin \mathrm{B}=\sin \mathrm{A}+\sin \mathrm{C}$
$\Rightarrow \quad 2 \sin B=\sin 2 C+\sin C \quad[\because A=2 C]$
$\begin{aligned} & \Rightarrow 2 \sin B=2 \sin \frac{3 C}{2} \cos \frac{C}{2} \\ & \text { also } A+B+C=\pi \Rightarrow 3 C+B=\pi \Rightarrow B=\pi-3 C \\ & \Rightarrow 2 \sin (\pi-3 C)=2 \sin \frac{3 C}{2} \cos \frac{C}{2} \\ & \Rightarrow 2 \sin 3 C=2 \sin \frac{3 C}{2} \cos \frac{C}{2} \\ & \Rightarrow 2 \sin \frac{3 C}{2} \cos \frac{3 C}{2}=\sin \frac{3 C}{2} \cos \frac{C}{2}\end{aligned}$
$\begin{aligned} & \Rightarrow \quad \sin \frac{3 \mathrm{C}}{2}\left(2 \cos \frac{3 \mathrm{C}}{2}-\cos \frac{\mathrm{C}}{2}\right) \\ & \Rightarrow \text { If } \sin \frac{3 C}{2}=0 \Rightarrow \frac{3 C}{2}=0 \text { or } \pi \Rightarrow C=0 \text { or } \frac{2 \pi}{3} \\ & \Rightarrow A=0 \text { or } \frac{4 \pi}{3} \text { (not possible) } \\ & \therefore 2 \cos \frac{3 C}{2}-\cos \frac{C}{2}=0 \\ & 2\left(4 \cos ^3 \frac{\mathrm{C}}{2}-3 \cos \frac{\mathrm{C}}{2}\right)-\cos \frac{\mathrm{C}}{2}=0 \\ & \cos \frac{\mathrm{C}}{2}\left(8 \cos ^2 \frac{\mathrm{C}}{2}-7\right)=0 \\ & \cos \frac{\mathrm{C}}{2}=0 \Rightarrow \frac{\mathrm{C}}{2}=90^{\circ} \Rightarrow \mathrm{C}=180^{\circ} \\ & \text { not possible } \\ & \therefore 8 \cos ^2 \frac{\mathrm{C}}{2}=7 \text { or } \cos ^2 \frac{\mathrm{C}}{2}=\frac{7}{8} \\ & \therefore \cos C=2\left(\frac{7}{8}\right)-1=\frac{3}{4} \\ & \sin \mathrm{C}=\sqrt{1-\left(\frac{3}{4}\right)^2}=\frac{\sqrt{7}}{4} \\ & \mathrm{~B}=\pi-3 \mathrm{C} \\ & \sin \mathrm{B}=\sin 3 \mathrm{C}=3 \sin \mathrm{C}-4 \sin ^3 \mathrm{C} \\ & =3\left(\frac{\sqrt{7}}{4}\right)-\frac{4 \times 7 \sqrt{7}}{64} \\ & =\frac{12 \sqrt{7}-7 \sqrt{7}}{16}=\frac{5 \sqrt{7}}{16} \\ & \therefore b: c=\sin \mathrm{B}: \sin \mathrm{C} \\ & =\frac{5 \sqrt{7}}{16}: \frac{\sqrt{7}}{4}=\frac{5}{4}: 1=5: 4 \\ & \end{aligned}$