Hint: The apparent depth of the dot depends on the combined refractive index of the medium.

Step 1: Find the apparent depth for medium 2.

Let the apparent depth be ${\mathrm{O}}_1$ for the object seen from ${\mathrm{\mu }}_2$, then;
                                   ${\mathrm{O}}_1=\frac{{\mathrm{\mu }}_2}{{\mathrm{\mu }}_1}\frac{\mathrm{h}}{3}$
Since, apparent depth = real depth /refractive index '$\mathrm{\mu }$'

Step 2: Find the apparent depth for medium 3.

Since, the image formed by Medium 1, ${\mathrm{O}}_1$ acts as an object for Medium 2.
If seen from ${\mathrm{\mu }}_3$, the apparent depth is ${\mathrm{O}}_2$.

Step 3: Find the apparent depth of the object.
Similarly, the image formed by Medium 2, ${\mathrm{O}}_2$ acts as an object for Medium 3.
                               ${\mathrm{O}}_2=\frac{{\mathrm{\mu }}_3}{{\mathrm{\mu }}_2}\left(\frac{\mathrm{h}}{3}+{\mathrm{O}}_1\right)$
$=\frac{{\mathrm{\mu }}_3}{{\mathrm{\mu }}_2}\left(\frac{\mathrm{h}}{3}+\frac{{\mathrm{\mu }}_2 \mathrm{h} }{{\mathrm{\mu }}_1 3}\right)=\frac{\mathrm{h}}{3}\left(\frac{{\mathrm{\mu }}_3}{{\mathrm{\mu }}_2}+\frac{{\mathrm{\mu }}_3}{{\mathrm{\mu }}_1}\right)$
Seen from outside, the apparent height is:
                                ${\mathrm{O}}_3=\frac{1}{{\mathrm{\mu }}_3}\left(\frac{\mathrm{h}}{3}+{\mathrm{O}}_2\right)=\frac{1}{{\mathrm{\mu }}_3}\left[\frac{\mathrm{h}}{3}+\frac{\mathrm{h}}{3}\left(\frac{{\mathrm{\mu }}_3}{{\mathrm{\mu }}_2}+\frac{{\mathrm{\mu }}_3}{{\mathrm{\mu }}_1}\right)\right]$
$=\frac{\mathrm{h}}{3}\left(\frac{1}{{\mathrm{\mu }}_1}+\frac{1 }{{\mathrm{\mu }}_2}+\frac{1}{{\mathrm{\mu }}_3}\right)$
This is the required expression of apparent depth.