A small object is placed $20cm$ from the first of a train of

three lenses with focal lengths, in order, of $10,15,$ and $20cm.$

The first two lenses are separated by $30cm$ and the last two

by $20cm.$ 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.