**Proof for the lens formula in a convex mirror:**

The lens formula is given by:

1/f = 1/u + 1/v

where:
- f is the focal length of the mirror
- u is the object distance (distance between the object and the mirror)
- v is the image distance (distance between the image and the mirror)

Let's prove this formula specifically for a convex mirror.

**Properties of a convex mirror:**

A convex mirror is a curved mirror that bulges outwards. The center of curvature (C) and the focal point (F) lie behind the mirror. The radius of curvature (R) is taken as negative for a convex mirror.

**Derivation:**

1. Consider an object placed in front of a convex mirror at a distance u.
2. The rays of light from the object diverge after reflection and appear to come from a virtual focal point (F) behind the mirror.
3. The distance from the mirror to the virtual focal point is the focal length (f) of the mirror.
4. From the properties of a convex mirror, the focal length (f) is half the radius of curvature (R). Therefore, f = -R/2.
5. The image formed by a convex mirror is always virtual, diminished, and erect.
6. The image distance (v) is negative for a virtual image formed by a convex mirror.
7. Using the sign convention, u is positive for an object placed in front of the mirror, and v is negative for a virtual image.
8. Substituting the values of f, u, and v into the lens formula:

1/(-R/2) = 1/u + 1/v

9. Simplifying the equation by taking the reciprocal of both sides:

-2/R = 1/u + 1/v

10. Rearranging the equation:

1/u + 1/v = -2/R

11. Since the focal length (f) is half the radius of curvature (R) for a convex mirror, we can substitute -2/R with -1/f:

1/u + 1/v = -1/f

Thus, the lens formula for a convex mirror is proven to be:

1/f = 1/u + 1/v