Solution

Originally, the source and its image are at the same position, which means that the source is at the center of curvature of the concave mirror. After some liquid is poured on the mirror, it becomes a thin convex lens with one flat surface whose focal length is $R/(n-1)$. A new image is formed after light rays from the source are refracted by the lens, reflected by the mirror, and refracted by the lens again. Thus the focal length $f$ of the compound lens-mirror-lens satisfies the equation

$$\frac{1}{f}=\frac{n-1}{R}+\frac{2}{R}+\frac{n-1}{R}=\frac{2n}{R}.$$

Since the new image is at the distance $l$ from the source (towards the mirror), we have

$$\frac{1}{f}=\frac{1}{R}+\frac{1}{R-l}=\frac{2R-l}{R(R-l)},$$

so $n=(2R-l)/[2(R-l)]$