a, b, c are in A.P. then
a = A - d, b = A, c = A + d ---(1)
where, A is the first term and D is the common difference of an A.P.
a + b + c = $\frac{3}{2}$ (given) ---(2)
Substituting equation (1) in (2):
A - d + A + A + d = $\frac{3}{2}$
3A = $\frac{3}{2}$
$\Rightarrow $ A = $\frac{1}{2}$ ---(3)
$a^2, b^2, c^2$ are in G.P.
$\Rightarrow (b^2)^2 = a^2c^2$ ---(4)
a = $\frac{1}{2} - d$, b = $\frac{1}{2}$, c = $\frac{1}{2}$ + d
Put these values and eq(3) in eq(4):
$\Rightarrow (\frac{1}{4})^2 = (\frac{1}{2}-d)^2(\frac{1}{2}+d)^2$
$\Rightarrow \frac{1}{16} = (\frac{1}{4}-d^2)^2$
$\Rightarrow \frac{1}{4}-d^2=\pm\frac{1}{4} $
$d^2=\frac{1}{2}$ (d=0 not possible)
$d=\pm\frac{1}{\sqrt{2}}$
$a=\frac{1}{2}\pm\frac{1}{\sqrt{2}}$
Correct option : D
