# New focal point of converging light through a concave lens

Suppose I pass rays of already converging light through a concave lens. The rays are equally centered on the lens such that the original focal point is through the lens and centered on the lens axis.

My understanding is that the lens will bend the incoming rays, such that they all focus on a new focal point closer to the lens. Is there a formula I can use to determine the location of the new focal point, given the focal point of the lens and the original focal point?

You start with just the convex lens so $u$ is the distance of the object and $v$ the distance to the image. Calculate $v$ using the lens equation:
$$\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$$
Now consider the second, concave, lens. Take the image from the first lens as the object, so the distance to the object is $u'$. Note that $u'$ will be a negative number because the convention is that to the left of the lens is positive and to the right of the lens is negative. Us the lens equation again (rememer $f$ is negative for concave lenses) with $u'$ as the object and you can calculate the position of the final image $v'$.