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

Let $a < b < c$ are three sides and $\theta$ is the smallest angle. Hence three angles are $\theta, \pi - 3\theta, 2\theta$
Given, $2b = a + c ⇒ 2sin B = sin A + sin C$ (using sine rule)
$⇒ 2\sin 3\theta = sin\theta + \sin 2\theta$
$⇒ 2$\left(3sin\theta - 4\sin^{3}\theta\right) = sin\theta\left(1 + 2cos\theta\right)$<br/>$⇒ 6 - 8\left(1 - \cos^{2}\theta\right) = 1 + 2cos\theta$<br/>$⇒ cos\theta = \dfrac{3}{4}, - \dfrac{1}{2}$($- \dfrac{1}{2}$is rejected since$\theta$is acute) <br/>$∴ a: b: c = sinA: sinB: sinC = sin\theta: \sin 3\theta: \sin 2\theta$<br/>$= 1: 3 - 4\sin^{2}\theta: 2cos\theta = 1: 4\cos^{2}\theta - 1: 2cos\theta$<br/>$= 4: 5: 6