Question

derive formula d/2(1/n1 + 1/n2)......d is the real depth, n1 and n2 are refractive index

Answer 1

let many medium which depth d1, d2, d3, ..... and refractive index n1 , n2 , n3 , n4 , ......

we know,
dapparent /dactual = R.I of medium of observer /R.I of medium of object
now use this concept here,

when observer placed in air then
dapparent = dactual/refractive index of medium of observer .

for this ,
dapp for all medium ,

dapp1 = d1/n1
dapp2 =d2/n2
dapp3 =d3/n3
...........
.........
dactual = d1 + d2 + d3 + .....dn

Nnet = dactual/dapp

Nnet = (d1 +d2 +d3 + ......dn)/(d1/n1+d2/n2+d3/n3 +......)

if d1 = d2 =d3 =.......dn

then,
Nnet = n/(1/n1 + 1/n2 + 1/n3 ++.....)

here net refactive index is Nnet