Kenneth Rosen Edition 7 Exercise 8.1 Question 21 (Page No. 511)

Question

• A. Find the recurrence relation satisfied by $R_{n},$ where $R_{n}$ is the number of regions that a plane is divided into by $n$ lines, if no two of the lines are parallel and no three of the lines go through the same point.
• B. Find $R_{n}$ using iteration.

Comments

Useful links: https://brainly.com/question/13058894

https://math.stackexchange.com/questions/263107/how-to-solve-this-recurrence-problem

Answer 1

One line divide a plane in 2 region and two unparalleled lines divide a plane in 4 regions.

So,

a) $R_{n}=R_{n-1}+n$

Suppose, there are $R_{n-1}$ parallel lines, which divides the plane in $R_{n-1}$ regions.

and 1 unparallel line , cut each parallel line in 1 point, which add n more region in the plane.