NMTC Practice Part 5

We solve this by recurrence relations.

Let $n$ lines divide the plane in $a_n$ parts.

Thus, when we draw $k^{th}$ line, there are already $a_{k-1}$ regions formed.

The $k^{th}$ line will intersect all the other $k-1$ lines, and this will add $k$ more regions (in the outer part, which you can call infinitely large regions.)

Thus we have $a_n=a_{n-1}+n$

Writing $a_{n-1} = a_{n-2} +n-1$ and so on, we have

$a_n= a_0 + \dfrac{n(n+1)}{2} = 1+ \dfrac{n(n+1)}{2}$

This is because 0 lines divide the plane in 1 part, the plane itself.

For $n=6$ , we have $a_6 = 1 + \dfrac{6\times 7}{2} = 1+21 =\boxed{22}$