smc

Note that, before adding the secant line, we have $2n$ regions, and, starting from the edge of the disk, every time the secant line intersects a radius or the other end of the disk, a new region is created. Thus, we desire to maximize the number of radii that the secant line passes through. The angle between any two adjacent radii must be $\frac{360^{\circ }}{2n}=\frac{180^{\circ }}{n}$. A given secant cannot pass through more than $180^{\circ }$ of the radii, and passing through exactly $180^{\circ }$ would imply that the secant must go through the center of the circle, and so it would not produce any new regions. Thus, to maximize the number of regions, the secant line must pass through $n$ radii (spanning $\frac{180}{n}(n-1)$ degrees) and thereby create $n+1$ new regions (one for each radius and one for exiting the disk). Thus, we have a maximum of $2n+n+1=\boxed{3n+1}$ regions.