A stack of k transparent slabs is arranged one on top of another,...

The apparent depth of an object viewed through a single slab of thickness $t and refractive index$\mu$is given by$\dfrac{t}{\mu}$.<br><br>For a stack of$k$slabs, the total apparent depth is the sum of the individual apparent depths:<br><br>Apparent depth$= \sum_{i=1}^{k} \dfrac{t_i}{\mu_i}$<br><br>If the entire assembly is replaced by a single equivalent slab of total thickness$\sum_{i=1}^{k} t_i$and equivalent refractive index$\mu_{eq}$, its apparent depth would be:<br><br>Equivalent apparent depth$= \dfrac{\sum_{i=1}^{k} t_i}{\mu_{eq}}$<br><br>For the image to form at the identical location, the total apparent depth must be equal to the equivalent apparent depth:<br><br>$\dfrac{\sum_{i=1}^{k} t_i}{\mu_{eq}} = \sum_{i=1}^{k} \dfrac{t_i}{\mu_i}$<br><br>Rearranging the above equation to solve for$\mu_{eq}$, we get:<br><br>$\mu_{eq} = \dfrac{\sum_{i=1}^{k} t_i}{\sum_{i=1}^{k} \left(\dfrac{t_i}{\mu_i}\right)}