An unpolarized light beam of intensity I_0 passes through a series...

Initial intensity of the unpolarized light is $I_0$.

When unpolarized light passes through the first polaroid $P_1$, the intensity becomes half. Thus, the intensity after $P_1$ is:

$I_1 = \dfrac{I_0}{2}$

For each subsequent polaroid $P_k$ ($k = 2, 3, \ldots, n$), the angle between its pass axis and the plane of polarization of the incident light is 60^{}^{\circ}.

According to Malus's Law, the intensity after passing through the $k$-th polaroid is:

I_k = I_{k-1} \cos^2(60^{}^{\circ})

Since \cos(60^{}^{\circ}) = \dfrac{1}{2}, we get:

$I_k = I_{k-1} \left(\dfrac{1}{2}\right)^2 = \dfrac{I_{k-1}}{4}$

Applying this relation successively for the remaining $(n-1)$ polaroids, the final intensity $I_n$ is:

$I_n = I_1 \left(\dfrac{1}{4}\right)^{n-1}$

Substituting $I_1 = \dfrac{I_0}{2}$:

$I_n = \dfrac{I_0}{2} \left(\dfrac{1}{2^2}\right)^{n-1} = \dfrac{I_0}{2 \cdot 2^{2n-2}} = \dfrac{I_0}{2^{2n-1}}$

Answer: $\dfrac{I_0}{2^{2n-1}}$