If the lengths of the sides of a triangle are in A.P.

$\mathrm{a}<\mathrm{b}<\mathrm{c}$ are in A.P.

$\angle \mathrm{C}=2 \angle \mathrm{A}$ (Given)

$\Rightarrow \sin C=\sin 2 A$

$\Rightarrow \sin C=2 \sin A \cdot \cos A$

$\Rightarrow \frac{\sin C}{\sin A}=2 \cos A$

$\Rightarrow \frac{\mathrm{c}}{\mathrm{a}}=2 \frac{\mathrm{b}^{2}+\mathrm{c}^{2}-\mathrm{a}^{2}}{2 \mathrm{bc}}$

put $\mathrm{a}=\mathrm{b}-\lambda, \mathrm{c}=\mathrm{b}+\lambda, \lambda>0$

$\Rightarrow \lambda=\frac{\mathrm{b}}{5}$

$\Rightarrow \mathrm{a}=\mathrm{b}-\frac{\mathrm{b}}{5}=\frac{4}{5} \mathrm{~b}, \mathrm{c}=\mathrm{b}+\frac{\mathrm{b}}{5}=\frac{6 \mathrm{~b}}{5}$

$\Rightarrow$ required ratio $=4: 5: 6$