With usual notations, in \(\Delta \mathrm{ABC}\), if \(\mathrm{b} \cos ^{2} \frac{\mathrm{C}}{2}+\mathrm{c} \cos ^{2} \frac{\mathrm{B}}{2}=\frac{3 \mathrm{a}}{2}\), then

Given \(\mathrm{b} \cos ^{2} \frac{\mathrm{C}}{2}+\mathrm{c} \cos ^{2} \frac{\mathrm{B}}{2}=\frac{3 \mathrm{a}}{2}\)
\(\therefore \quad \mathrm{b}\left(\frac{1+\cos \mathrm{C}}{2}\right)+\mathrm{c}\left(\frac{1+\cos \mathrm{B}}{2}\right)=\frac{3 \mathrm{a}}{2}\)
\(\therefore \mathrm{b}+\mathrm{b} \cos \mathrm{C}+\mathrm{c}+\mathrm{C} \cos \mathrm{B}=3 \mathrm{a} \Rightarrow(\mathrm{b} \cos \mathrm{C}~+\) \(\mathrm{c} \cos \mathrm{B})+\mathrm{b}+\mathrm{c}=3 \mathrm{a}\)
\(\quad \mathrm{a}+\mathrm{b}+\mathrm{c}=3 \mathrm{a} \Rightarrow \mathrm{b}+\mathrm{c}=2 \mathrm{a} \Rightarrow \mathrm{a}=\frac{\mathrm{b}+\mathrm{c}}{2}\)
\(\therefore \quad \mathrm{b}, \mathrm{a}, \mathrm{c}\) are in A.P.