To explain the recurrence relation $R_n = n + R_{n-1}, R_0 = 1$, let’s break down the problem step by step.

Problem Overview

We need to determine the number of regions ($R_n$) that a plane is divided into by $n$ lines under the conditions:

1. No two lines are parallel.
2. No three lines are concurrent (i.e., no three lines pass through a single point).

Step-by-Step Explanation

1. Base Case:

   • When $n = 0$ (no lines), the entire plane is undivided and thus forms a single region: $R_0 = 1.$

2. Adding the First Line:

   • Adding one line to the plane divides the plane into two regions: $R_1 = 2.$

3. Adding the $n$-th Line:

   • When we add the $n$-th line to the existing $n-1$ lines, this new line intersects all $n-1$ lines already present. Since no two lines are parallel and no three lines meet at a single point, the $n$-th line intersects each of the $n-1$ lines at a unique point.
   • These intersection points divide the $n$-th line into $n$ segments.
   • Each of these segments creates a new region in the plane.

   Hence, the $n$-th line adds $n$ new regions to the plane.

4. Recurrence Relation:

   • The total number of regions after adding the $n$-th line is therefore: $R_n = R_{n-1} + n.$ This is because $R_{n-1}$ accounts for all regions created by the previous $n-1$ lines, and $n$ new regions are added by the $n$-th line.

5. Initial Condition:

   • As stated earlier, when $n = 0$, the plane has only one region: $R_0 = 1.$

Final Recurrence Relation

Thus, the recurrence relation is: $R_n = R_{n-1} + n, \quad R_0 = 1.$

Verification

To confirm, let’s compute the first few terms:

• $R_0 = 1$.
• $R_1 = R_0 + 1 = 1 + 1 = 2$.
• $R_2 = R_1 + 2 = 2 + 2 = 4$.
• $R_3 = R_2 + 3 = 4 + 3 = 7$.
• $R_4 = R_3 + 4 = 7 + 4 = 11$.

These match the expected number of regions for $n = 0, 1, 2, 3,$ and $4$.

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Key Insight:

The recurrence captures the incremental increase in regions caused by the addition of each new line. This systematic addition arises from the geometric constraints of the problem.

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Would you like me to explain further details or help with a related concept? Here are some questions to explore further:

1. How can you derive a closed-form expression for $R_n$ instead of a recurrence relation?
2. Can we generalize this problem to $k$-dimensional space (e.g., planes in 3D space)?
3. What if the lines are parallel? How does the recurrence change?
4. How does the number of intersection points relate to the number of regions?
5. How would the recurrence change if three lines were allowed to meet at a point?

Tip: Drawing the regions for small values of $n$ (e.g., $n = 1, 2, 3$) can help build intuition for recurrence relations in geometry problems.