Two thin lenses of focal lengths `f_(1)` and `f_(2)` are in contact. The focal length of this combination is

Explanation

Focal length of combination of two lenses placed in contact with each other:

Consider an object placed at a distance u on principal axis of the lens. Rays of light starting from O from the image I.
`:. (1)/(v_(1))-(1)/(u)=(1)/(f_(1))` ...(1)
Place the lens B in contact with A. The image `I_(1)`, will serve as virtual object and forms a real image I.
`:.` Again applying lens formula,
`-(1)/(mu_(1))+(1)/(v)=(1)/(f_(2))` ...(2)
Adding (1) and (2), we get
`(1)/(v)-(1)/(u)=(1)/(f_(1))+(1)/(f_(2))` ...(3)
But `(1)/(v)-(1)/(u)=(1)/(f)`, where f is the focal length of combination.
So Eq. (3) becomes
`(1)/(f)=(1)/(f_(1))+(1)/(f_(2))`