Let $R_n$ denote the number of regions into which the plane is divided by n lines. Assume that each pair of lines meets in a point, but that no three lines meet in a point. Derive a recurrence relation for the sequence $R_1$, $R_2$,...
I understand that $R_1$, $R_2$ means the number of regions in which the plane is divided
$R_1$ is the plane divided into 1 region, $R_2$ into 2 regions
$R_1$ has one line, $R_2$ has 2 lines... Would I have to use the formulas to find points in a plane? but I don't quite understand it, how many formulas would it be
I would greatly appreciate the help, I have to expose this exercise to raise points on this subject.
As the question itself says:
So, it means that if you draw out the cases for a few of them, you should be able to see a pattern, you have to recognise/identify that.
So, you start by drawing lines, starting from $n=1,2\cdots:$
In last one, $n=4$, you have to be especially careful so that no three lines meet in a point.
As you can see, we observe a pattern coming out of it as:
$R_n=\sum n +1=\frac {n(n+1)}2+1$
And similarly the relation:
$R_n=R_{n-1}+n$ where $n\geq2$ and $R_1=2$.