Help: can you help me come up with a formula for the number of regions?

Question

The "missing" region

Mark a number of points unevenly round a circle and join every point to every other point. How many regions has the circle been divided into? Can you find a formula to predict the number of regions Y given the number of points X?

The diagram alongside shows that there are Y = 8 regions created with X = 4 points. The table below gives some more values which you can check if you like!
Points X 2 3 4 5 6 7
Regions Y 2 4 8 16 31 57
Differences D 2 4 8 15 26

Can you find a (polynomial) function for Y(X) ?


the diagram is within the pdf linked: http://www.itemaths.org.uk/using-ict/graphical-calculators-ks3.pdf

Answer 1

With three points you form a triangle, with four points a quadrilateral. Regions "outside" the polygon are equal to the number of sides (the number of points). The number of regions inside the polygon is the interesting number.

For two points, there are no regions "inside" the polygon.
For three points, there is one region inside the polygon.
For four points, there are four regions inside.