Derive the expression for effective focal length of two thin lenses kept in contact.

Explanation

In the above figure
OP= u = object distance
PI = V = image distance due to the combination
`PI_(1) = `v_(1)` = image distance due to first lens. For the lens
For the lens `A,-(1)/(v)-(1)/(v_(1))=(1)/(f_(2))`………….(2)
`Adding equations (1) and (2)
`(1)/(v)-(1)/(u)=(1)/(f_(1))+(1)/(f_(2))=(1)/(f) (because (1)/(v)-(1)/(u)=(1)/(f))`
`(1)/(f)=(1)/(f_(1))+(1)/(f_(2))`