1. A single line divides a plane into two regions. Two lines can divide a plane into four regions, and
three lines can divide a plane into seven regions.
A). Derive an explicit formula for the number of regions that can be obtained with n lines.
B). Prove your claim fro A).
Solution
a)
As a recurring way, consider you have (n-1) lines. Then a new line (n-th line) have to cross over
the others by this way: Region-Line-Region-Line-...-Line-Region. We know that is passes (n-1)
lines, and each line it passes a region will be passed too. And of course it starts with a region. So
we have (n-1)+1=n number of regions which the line pass them. Moving through them by the
line, split each of them into two regions. So all together we have new n regions. Thus the
recurring formula will be:
P(n) = P(n-1) + n (Where n is an integer & n>=2 And also P(1)=2)
b)
we know that, Two lines can divide a plane into four regions,
from the formula,
=> p(2) = p(1) + 2 Â = 2 + 2 = 4........proving the formula