A small object is placed 20 cm from the first of a train of three lenses with focal lengths , in order , of 10 , 15 , and 20 cm . The first two lenses are separated by 30 cm and the last two by 20 cm . Calculate the final image position relative to the last lens and its linear magnification relative to the original object when (a) all three lenses are positive ;(b) the middle lens is negative ;(c) the first and last lenses are negative . Provide ray diagrams for each case .

A small object is placed 20 cm from the first of a train of three lenses with focal lengths , in order , of 10 , 15 , and 20 cm . The first two lenses are separated by 30 cm and the last two by 20 cm . Calculate the final image position relative to the last lens and its linear magnification relative to the original object when (a) all three lenses are positive ;(b) the middle lens is negative ;(c) the first and last lenses are negative . Provide ray diagrams for each case .