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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.

insipidintegrator
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Lisstalik
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  • Hint: Let $R_n$ be the number of segments for n lines. Introduce a new line. How many extra parts does it introduce into the plane? – insipidintegrator Aug 22 '22 at 05:29
  • @insipidintegrator it would be that the plane is divided into even parts? – Lisstalik Aug 22 '22 at 05:52
  • Another hint: the recurrence should be $a_n =a_{n-1}+n$. – insipidintegrator Aug 22 '22 at 06:12
  • You could also draw out the case for 1,2,3,4 lines. The no of regions follows an obvious $1+\sum_{i=1}^ni$ from there. – Cathedral Aug 22 '22 at 06:17
  • @insipidintegrator a user with a high reputation answered me and then deleted, said: Suppose there are n lines on the plane. Now take one particular line, it divides the plane into two halfplanes. Now the n−1 other lines divide each of the half planes into Rn−1 regions, thus we have: Rn=2Rn-1

    so would be wrong?

     – Lisstalik Aug 22 '22 at 06:36
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    @Lisstalik That can be refuted by simple calculation of some initial values as was pointed out in their comments: This would imply that $R_1=2,R_2=4, R_3=8…$ but you can notice (by hand) that $R_3=7$. Take, for eg. the coordinate axes and the line x+y=1. – insipidintegrator Aug 22 '22 at 06:38
  • @Lisstalik "R1 is the plane divided into 1 region" $;-;$ How so? Dividing something into one piece is a contradiction of terms. Draw a line in the plane and count the regions it divides it into - that's R1 and it's quite obviously not 1. – dxiv Aug 22 '22 at 06:50

1 Answers1

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As the question itself says:

Derive a recurrence relation for the sequence R1, R2,...

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 $n=1,2\cdots:$

n #Regions $\sum n$
1 2 1
2 4 3
3 7 6
4 11 10

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 $R_1=2$.

InanimateBeing
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