Into how many regions is a circle divided by the lines connecting all pairs of n points on the circumference?

Question

Assume that the n points are spaced so that no 3 line segments have a common intersection point inside the circle.
To start, if the answer is denoted R(n), note that R(2)=2, R(3)=4, R(4)=8

NOTE TO Helmut:
You haven't performed any induction.
FYI: R(11) = 386

Answer 1

i derived a recursive formula as:

f(n) = f(n-1) + (n-1)(n^2-5n+12)/6

eg: f(3) = 4 as we know

f(4) = 4 + 3*8/6 = 8
f(5) = 8 + 4*12/6 = 16
f(6) = 16 + 5*18/6 = 31 etc


hint: the above formula was derived from the relation:

s = (n-3) + 2 (n-4) + 3 (n-5) +...+ (n-2) (n-n)

so that f(n) = f(n-1) + s

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Though this is recursive, one can easily derive an absolute expression....

Answer 2

By induction,
if r(2)=2, R(3)=4, R(4)=8, R(5)=16, then
R(n)=2^(n-1)