Derive the equation for effective focal length for lenses in out of contact.

Explanation

Consider a two thin lenses in contact. In the absence of second lens `L_(2) "the fist lens" L_(1)` will form a real image I.. Using thin lens formula.
`(f)/(f_(1))=(1)/(v)-(1)/(u)`
The image I. acts as a virtual object (u = v.) for the seond lens `L_(2)` which finally forms its real image I at distance v. Thus
`(1)/(f_(2))=(1)/(v)-(1)/(v.)`
Adding equation (1) and (2) we ge t,
`(1)/(f_(1))+(1)/(f_(2))=(1)/(v)-(1)/(u)`
For the combination of thin lenses in contact, if f is the equivalent focal lenght , then `(1)/(f_(1))+(1)/(f_(2))=(1)/(v)-(1)/(u)` Form equation (3) and (4), the effective focal lenght of lenses in contace.
`(1)/(f)=(1)/(f_(1))+(1)/(f_(2))`