# ASAP: Identifying the series (1 Viewer)

#### amdspotter

##### Member
Acknowledging the fact that no lines are parallel and that no three lines are collinear, how would you find the number of enclosed regions formed from these lines. what would the series be?
i.e 1 line will create 0 enclosed
2 lines will create 0 enclosed regions
3 lines will create 1 enclosed region
@icycledough @CM_Tutor , Anyone - any help is appreciated.

#### CM_Tutor

##### Moderator
Moderator
Here is a related problem:

The solution for part (i) is:

Based on this solution, we hypothesise that the pattern is $\bg_white s_n = s_{n-1} + n,\ \text{for integers n \geqslant 2 with s_1 = 2.$

This can be shown to give $\bg_white s_n = \cfrac{1}{2}(n^2 + n + 2)$ by recognising that we can shown that $\bg_white s_n = 1 + \big(1 + 2 + 3 + ... + n\big)$ and summing the AP. This formula can be proven to be correct by mathematical induction.

The problem you are asking about is this problem with the non-closed regions removed. See if you can solve it from here.

Last edited:

#### amdspotter

##### Member
Here is a related problem:
View attachment 34330

The solution for part (i) is:
View attachment 34331

Based on this solution, we hypothesise that the pattern is $\bg_white s_n = s_{n-1} + n,\ \text{for integers n \geqslant 2 with s_1 = 2.$

This can be shown to give $\bg_white s_n = \cfrac{1}{2}(n^2 + n + 2)$ by recognising that we can shown that $\bg_white s_n = 1 + \big(1 + 2 + 3 + ... + n\big)$ and summing the AP. This formula can be proven to be correct by mathematical induction.

The problem you are asking about is this problem with the non-closed regions removed. See if you can solve it from here.
@CM_Tutor I tried to solve it for my problem but wasn't able to really do it, could you if possibly explain it how to it with reference to the problem I am trying to solve

#### Modern4DaBois

##### Member
Very interesting question you got there @amdspotter

#### CM_Tutor

##### Moderator
Moderator
The pattern of non-enclosed regions follows a simple pattern: 2, 4, 6, 8, 10, ...

Imagine a 2D plane with some number of intersecting lines. Now, imagine yourself riding a new line in from infinity. The first line you meet will create a new non-enclosed region. The next will create a new enclosed region, as will each subsequent line it meets. After crossing the final line and continuing off to infinity, it creates a new non-enclosed region... which is why each new line adds two new non-enclosed regions.

So, the series of enclosed regions, $\bg_white e_n$, follows $\bg_white e_n = s_n - 2n$ for $\bg_white n > 2$. That is,

\bg_white \begin{align*} e_3 &= s_3 - 2(3) = 7 - 6 = 1 \\ e_4 &= s_4 - 2(4) = 11 - 8 = 3 \\ e_5 &= s_5 - 2(5) = 16 - 10 = 6 \end{align*}

It follows that

$\bg_white e_n = \cfrac{1}{2}(n^2 + n + 2) - 2n = \cfrac{1}{2}(n^2 - 3n + 2) = \cfrac{1}{2}(n - 1)(n - 2)$

which also works for the cases $\bg_white e_1 = 0$ and $\bg_white e_2 = 0$.

#### CM_Tutor

##### Moderator
Moderator
Undeleted as I got an edit conflict in posting the solution.

#### amdspotter

##### Member
The pattern of non-enclosed regions follows a simple pattern: 2, 4, 6, 8, 10, ...

Imagine a 2D plane with some number of intersecting lines. Now, imagine yourself riding a new line in from infinity. The first line you meet will create a new non-enclosed region. The next will create a new enclosed region, as will each subsequent line it meets. After crossing the final line and continuing off to infinity, it creates a new non-enclosed region... which is why each new line adds two new non-enclosed regions.

So, the series of enclosed regions, $\bg_white e_n$, follows $\bg_white e_n = s_n - 2n$ for $\bg_white n > 2$. That is,

\bg_white \begin{align*} e_3 &= s_3 - 2(3) = 7 - 6 = 1 \\ e_4 &= s_4 - 2(4) = 11 - 8 = 3 \\ e_5 &= s_5 - 2(5) = 16 - 10 = 6 \end{align*}

It follows that

$\bg_white e_n = \cfrac{1}{2}(n^2 + n + 2) - 2n = \cfrac{1}{2}(n^2 - 3n + 2) = \cfrac{1}{2}(n - 1)(n - 2)$

which also works for the cases $\bg_white e_1 = 0$ and $\bg_white e_2 = 0$.
Hey there, what exactly is "s" in this?

#### CM_Tutor

##### Moderator
Moderator
$\bg_white s_n$ is defined in my first post in this thread, the number of distinct segments of a plane split by $\bg_white n$ all non-parallel lines.

#### cossine

##### Active Member
Hey there, what exactly is "s" in this?
s sub n is the number of regions.