When (n-1) polaroids are placed between, two polaroids. Total number of polaroids becomes (n-1+1+1) = (n+1). The axis of all the polaroids are equally spaced. If x is angle between the axis of the two consecutive polaroids then:
[tex]{:\implies \quad \sf x+x+x+\cdots \cdots n\:\:times=\dfrac{\pi}{2}}[/tex]
[tex]{:\implies \quad \sf nx=\dfrac{\pi}{2}}[/tex]
[tex]{:\implies \quad \sf x=\dfrac{\pi}{2n}}[/tex]
Now, by Malus law, we know that the intensity of light on passing through a pair of Polaroid is proportional to cos²(x). Before the light passes out of the last polaroid, this change in intensity will be repeated n times. If [tex]{I_0}[/tex] is intensity of the incident light and [tex]{I}[/tex], is the intensity of light after passing through all the polaroids, then mathematically from Malus law:
[tex]{:\implies \quad I=I_{0}\sf \cos^{2n}(x)}[/tex]
[tex]{:\implies \quad I=I_{0}\sf \cos^{2n}\bigg(\dfrac{\pi}{2n}\bigg)}[/tex]
When, n will be very large, the angle x → 0, so that the whole cosine expression → 1. So, when n will be very much large, [tex]{I}[/tex] will approach [tex]{I_0}[/tex]
