Lines Partitioning a Plane

We can find a general formula for the number of regions when a plane
is divided by n lines.

Suppose the number of regions is given by f(n) when there are n lines
drawn on the plane. Now draw one more line cutting all the other n
lines. There are n points on the additional line and so this line must
traverse n+1 of the available f(n) regions, dividing each into 2
parts. It therefore adds n+1 more regions to those present. Therefore:

f(n+1) = f(n) + n + 1

We can write f(n+1) - f(n) = n + 1, and now form a succession of
equations as follows:

f(1) - f(0) = 1
f(2) - f(1) = 2
f(3) - f(2) = 3
...................
...................
f(n-1) - f(n-2) = n-1
f(n) - f(n-1) = n
------------------------- adding all the equations, we get
f(n) - f(0) = SUM(1 to n) cancellations between lines,

f(n) - 1 = n(n+1)/2

f(n) = n(n+1)/2 + 1

Check if this is correct: n = 0 gives f(0) = 1
n = 1 gives f(1) = 2
n = 2 gives f(2) = 4,
which they are.

With n = 5, we get 5 * 6/2 + 1 = 16 regions.

So 5 lines will divide the plane into 16 regions.