Find the recurrence relation satisfied by $R_n$, where $R_n$ is the number of regions into which the surface of a sphere is divided by n great circles (which are the intersections of the sphere and planes passing through the center of the sphere), if no three of the great circles go through the same point
From: Rosen Discrete Mathematics textbook
My attempt: I was unable to visualize these greater circles for values of $n \geq 4$. So, I launched GeoGebra and made the following plot:
My initial planes were $x=y$, $x=-y$, and the $z=0$ plane. Now, I added a fourth yellow plane $y=z$. However, I still find it hard to understand how to calculate what number of new sections are formed. I am able to calculate the number of new sections in this specific case, but, I am unable to arrive at a general formula for the new number of sections, through recurrence.
I visited the following website - Slader - which happens to have a solution, where they mention the addition of $2(n-1)$ regions to the existing regions, after the addition of the $n$-th plane. However, I am unable to understand why exactly that is true.