A problem is similar to (see my solution).
You are adding new point on the circle and drawing new chords. For each chord you count intersections with old chords and add 1, this gives you the number of new regions. This is how you arrive at the formula. Last step is to simplify it.
But to be honest I just guessed the solution
I took the formula from above mentioned problem and made an adjustment for that extra 1s. Few tries and it worked.
Edit: Guessing helped me only in the last step (simplification). I needed non-simplified formula to verify guessed one.