Step 1: Approach
Let the angles be $A=B-d$, $B$, $C=B+d$ and apply the sine rule.
Step 2: Middle angle
$A+B+C=3B=180^\circ$, so $B=60^\circ$.
Step 3: Ratio
$\dfrac{\sin B}{\sin C}=\dfrac{b}{c}=\sqrt{\dfrac32}$, so $\sin C=\sin60^\circ\sqrt{\dfrac23}=\dfrac{\sqrt3}{2}\cdot\dfrac{\sqrt2}{\sqrt3}=\dfrac{1}{\sqrt2}$.
Step 4: Angles
$C=45^\circ$ is the only possible value ($135^\circ$ would push the angle sum past $180^\circ$). Then $A=180^\circ-60^\circ-45^\circ=75^\circ$. The angles $45^\circ, 60^\circ, 75^\circ$ are in A.P. with difference $15^\circ$, so option (D) is correct.
Final Answer:
The angles are 45, 60 and 75 degrees, so angle A is 75 degrees, option (D).
\[ \boxed{75^{\circ}} \]