Take a sphere and draw on it a great circle (a great circle is a circle whose centre is the centre of the sphere). There are two regions created. Here, I am referring to regions on the surface of the sphere. Now draw another great circle: there are four regions. Now draw a third, not passing through the points of intersection of the first two. How many regions?
Here's the general question: How many regions are created by n great circles, no three concurrent, drawn on the surface of the sphere?