Different geometry problem

When I posted the first one I meant to post a slightly different problem. Here it is:

Draw a circle. Draw n points on the circumference. Join up the points with straight lines.
How many regions can you divide the circle into?

this would be the same problem as the other one, except that the
number of lines has to be calculated first: the combination of n
elements in groups of two: n x (n-1) / 2

And we substitute that into the formula of the other problem

number of regions = 1 + [N x (N+1)] / 2 (the other problem)
with N= n x (n-1) / 2
gives : 1 + [ n x (n-1) x ( n² - n + 2)] / 8

lets try with
2 points: 1 line, 2 regions
3 points: 3 lines 7 regions
4 points: 6 lines, 22 regions
etc.....

or am I talking nonsens (again)?

sin.

Your first answer was correct, but this answer isn't, I'm afraid. You can't make 7 regions with
just 3 points. Hint: the lines aren't arbitrary, as they were in the first problem.

you are right (nothing to be afraid of, LOL). I was a bit naive. There
are two important differences with the first problem:

1) In the first problem I could imagine the circle outside ALL
intersections. So the area's between intersections laying outside the
circle have to be deducted.

2) in the first problem, all lines could have intersected at different
points; Here there are a lot of coinciding intersections.

Too complicated to answer just like that (at least starting from this
angle). Back to the study .... Anyone else?

I find the easiest way to think about this is by induction. Put a few points down, and see
what hapens when you add one more point.

2 points = 2 regions.
3 points = 4 regions.
4 points = 8 regions.
5 points = 16 regions.
6 points = 30 regions?? Am I counting wrong?

For 6 I can make more than 30. Don't put the points in a regular hexagon.

with irregular placement I count 31 regions - still not the expected 32.

for n is odd, the terms follow the pattern
n,n+1,3n+1,5n+1 etc (because you always have that irritating central region)
and for n is even the terms follow the pattern
n, 2n, 4n, 6n, 8n etc.
its a compound series, but thats as far as my memory will get me. i tried to express it in terms of n but i get all fuzzy and irritated. there must be a mathematician or two on this site who see these things when they watch the sky...