We have seen that if to a circle spots are added then the resutling regions formed are as follows.

No.of spots are 1, 2 3 4 5 6

No. of regionsl 1, 2 4 8 16 31

The 1st 5 terms are in G.P but from the 6th T6 =31 and not 32 as expected.

We must note that any point where 3 or more lines intersect has to be counted as a zero area region [Thanks Paul]

Concurrent lines like the spots themselves are not to be counted.

Quest: What are the next two terms?

This is sufficient data to to work out the nth term Tn.

Derive formula for Tn.

Hint: There are at least three different ways in which Tn can be expressed.

Mally