f(n) denotes the number of regions in the circle with n lines in it.
As you say, by drawing one more line there will be n points on that
line. In other words, of the f(n) regions the line will traverse n+1 of
them. Each region gets split into two so n+1 regions are added as a
result of this extra line.
So before we had f(n), now we have f(n+1) and this extra line has
given us n+1 extra regions.
Therefore, f(n+1) = f(n) + n+1
Now the cunning bit...
Rearranging the above line will give f(n+1) - f(n) = n+1
Consider what this is saying (sorry for all the dots, they are necessary
to line up the columns, if I leave spaces the forum software scrunches
f(1) - f(0) = 1 (since here n = 0 so n+1 = 0+1 = 1)
f(2) - f(1) = 2 (since here n = 1 so n+1 = 1+1 = 2)
f(3) - f(2) = 3 (since...etc)
f(4) - f(5) = 4
and so on up to:-
f(n-1) - f(n-2) = n-1
f(n) - f(n-1) = n
Add up the columns (note how the top left to bottom right diagonal on
the left hand side of the equals sign cancels, this is what Acolyte
mentioned in one of his posts where all the terms cancel leaving just
two, (usually) just the first one and the last one).
We get f(n) - f(0) = 1+2+3+4+...+(n-1)+n
The right hand side is simply the sum of the first n integers!!! This is
where it ties in with the other thread involving Gauss and arithmetic
progressions!! I love maths! We can use the formula Hotpawn and
Gilbert gave in that thread (the proof of which follows the same
pattern as the explanation I gave Jon with n replacing 100).
So, we have:-
f(n) - f(0) = n(n+1)/2
We know f(0) to be 1 so...
f(n) - 1 = n(n+1)/2
Which, adding one to both sides, is the formula that Gilbert arrived at
by intuition and testing the first few cases.
f(n) = n(n+1)/2 + 1