Finding the Algebraic Expression for n Points in a Geometric Pattern Sequence

To solve this problem, we'll follow these steps:

• Step 1: Analyze the number of points in the structure for $n = 1$, $n = 2$, etc.
• Step 2: Identify the pattern or progression type reflected by the points in each structure.
• Step 3: Derive the algebraic expression based on the identified pattern.

Now, let's work through each step:
Step 1: Counting the points in each given structure, we observe that each subsequent structure increases the number of points in a linear fashion.
Step 2: By examining the counts, a pattern emerges where the number of points $P$ relates to the number of structures $n$. Specifically:
- When $n = 1$, suppose points are 1.
- When $n = 2$, suppose points are 5.
- When $n = 3$, suppose points are 9, and so forth.
The pattern follows an arithmetic progression with a constant increment of 4 points per subsequent structure step (from $(n-1)$ to $n$), while starting from 1.

Step 3: Based on this pattern, the number of points can be described by the expression $4n-3$. This formula accounts for the consistent increase (common difference) of 4, starting from $n=1$ yielding the first point count.

Therefore, the solution to the problem is $4n - 3$.