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Find the recurrence relation satisfied by rn, where rn 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.

Respuesta :

1. Take a look at the pictures attached.

2. 

i) the first line divides the plane into 2 regions 

ii) the second line adds 2 more regions so we have 4 in total.

iii) the third line adds 3 more regions, so 4+3=7 regions

iv) the fourth line adds 4 more regions.

so the [tex] n^{th} [/tex] line adds n more regions to the ones created by the previous n-1 lines.

3. 

[tex]r_1=2[/tex]

[tex]r_2=2+2=4[/tex]

[tex]r_3=4+3=7[/tex]

[tex]r_4=7+4=11[/tex]

[tex]r_n=r_n_-_1+n[/tex]

So the recurrence relation is

[tex]r_1=2[/tex]
[tex]r_n=r_n_-_1+n[/tex]
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