Suppose we draw n straight lines on the plane so that every pair of
lines intersects (but no 3 lines intersect at a common point). Into
how many regions do these n lines divide the plane?
With n = 1 we divide the plane into 2 regions. With n = 2 we have 4
regions; with n = 3 we get 7 regions. A fourth line will meet the
other 3 lines in 3 points and so traverse 4 regions, dividing them
into two parts and adding 4 new regions. In general the nth line will
add n new regions:
u(1) = 2
u(2) = 4
u(3) = 7
u(4) = 11
And so on, where u(n) = number of regions with n lines.
We get the recurrence relationship:
u(n+1) = u(n) + (n+1)
We get the following chain of equations:
u(n) - u(n-1) = n
u(n-1) - u(n-2) = n-1
u(n-2) - u(n-3) = n-2
......................
........................
u(4) - u(3) = 4
u(3) - u(2) = 3
u(2) - u(1) = 2
--------------------------
u(n) - u(1) = 2 + 3 + 4 + ..... + (n-1) + n
which we find by summing the equations.
All other terms on the left cancel between rows. We are left with:
u(n) = u(1) + 2 + 3 + 4 + ....+ n and u(1) = 2
Thus:
u(n) = 1 + [1+2+3+4+...+n]
= 1 + n(n+1)/2
= (2 + n^2 + n)/2
So:
u(n) = (n^2+n+2)/2
If you allow parallel lines and more than 2 lines to intersect at a
point, the answer becomes undefined.