A Mathematical Induction Problem
From my textbook
Give this problem a try before jumping in for the solution!
Solution
This problem is an application of mathematical induction. We will first consider the case when n = 1, that is when the region is divided by one line.
As shown above, It is certainly true that a single line divides a plane into two regions.
The statement is true for n = 1.
We now assume the statement is true for n = k, that is if k non-parallel lines divide the plane into r regions, then
Consider k + 1 non-parallel lines dividing the region
For the lower bound: when n = k, the plane is divided into 2k regions when all k lines intersect at a single point
When n = k + 1 and all k + 1 lines intersect at a single point, there are two more regions created.
For the upper bound, when n = k, the plane is divided into 1/2(k² + k + 2) regions when only two lines intersect at any point
When n = k + 1 and only two lines intersect at any point, k + 1 more regions are created.
For example, when n = 2
4 regions are created if only 2 lines intersect at any single point
When n = 3
7 regions are created when only two lines intersect at any single point, which means there are 3 more regions when the third line is added. The same reasoning applies to n = 4, 5, 6 and so on.
Therefore, for n = k + 1
Hence
If the statement holds true for n = k, it holds true for n = k + 1
Therefore the statement holds true for all positive integers n.
Well done!
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Happy reading,
Barry 🍩