An arrangement consists of three conducting concentric spherical...

The given arrangement can be considered as two spherical capacitors connected in series.

The first capacitor $C_1$ is formed by the inner shell of radius $a$ and the middle shell of radius $b$. The space between them is filled with a dielectric of constant $K$. Its capacitance is:
$C_1 = \dfrac{4\pi\varepsilon_0 K a b}{b - a}$

The second capacitor $C_2$ is formed by the middle shell of radius $b$ and the outer shell of radius $c$. The space between them is vacuum (or air). Its capacitance is:
$C_2 = \dfrac{4\pi\varepsilon_0 b c}{c - b}$

Since the terminals $A$ and $B$ are connected to the innermost and outermost shells respectively, and the middle shell is isolated, the two capacitors are in series. The equivalent capacitance $C_{eq}$ is given by:
$\dfrac{1}{C_{eq}} = \dfrac{1}{C_1} + \dfrac{1}{C_2}$

Substituting the values of $C_1$ and $C_2$:
$\dfrac{1}{C_{eq}} = \dfrac{b - a}{4\pi\varepsilon_0 K a b} + \dfrac{c - b}{4\pi\varepsilon_0 b c}$

Taking the common denominator:
$\dfrac{1}{C_{eq}} = \dfrac{c(b - a) + K a(c - b)}{4\pi\varepsilon_0 K a b c}$

Inverting this expression gives the equivalent capacitance:
$C_{eq} = \dfrac{4\pi\varepsilon_0 K a b c}{K a(c - b) + c(b - a)}$