The task is to make number sequences based on properties of regular polygons, such as the number of sides, sum of interior angles, sum of exterior angles, and so on. Here's how to analyze the sequences:

Step 1: Analyze Regular Polygons

The sequence starts with equilateral triangles (3 sides), squares (4 sides), regular pentagons (5 sides), and so on. For each polygon:

1. Number of sides: This is the simplest sequence: $3, 4, 5, 6, \dots$.

2. Sum of interior angles: The sum of interior angles of a polygon is calculated using: $\text{Sum of interior angles} = (n-2) \cdot 180^\circ$ where $n$ is the number of sides.

   For $n = 3, 4, 5, 6$:

   • Triangle ($n=3$): $(3-2) \cdot 180 = 180^\circ$
   • Square ($n=4$): $(4-2) \cdot 180 = 360^\circ$
   • Pentagon ($n=5$): $(5-2) \cdot 180 = 540^\circ$
   • Hexagon ($n=6$): $(6-2) \cdot 180 = 720^\circ$
   • Sequence: $180, 360, 540, 720, \dots$.

3. Sum of exterior angles: For any regular polygon, the sum of exterior angles is always $360^\circ$, regardless of the number of sides.

4. One interior angle (regular polygons): Each interior angle is: $\text{One interior angle} = \frac{\text{Sum of interior angles}}{n} = \frac{(n-2) \cdot 180}{n}$ For $n = 3, 4, 5, 6$:

   • Triangle: $\frac{(3-2) \cdot 180}{3} = 60^\circ$
   • Square: $\frac{(4-2) \cdot 180}{4} = 90^\circ$
   • Pentagon: $\frac{(5-2) \cdot 180}{5} = 108^\circ$
   • Hexagon: $\frac{(6-2) \cdot 180}{6} = 120^\circ$
   • Sequence: $60, 90, 108, 120, \dots$.

5. One exterior angle: Each exterior angle is: $\text{One exterior angle} = \frac{360}{n}$ For $n = 3, 4, 5, 6$:

   • Triangle: $\frac{360}{3} = 120^\circ$
   • Square: $\frac{360}{4} = 90^\circ$
   • Pentagon: $\frac{360}{5} = 72^\circ$
   • Hexagon: $\frac{360}{6} = 60^\circ$
   • Sequence: $120, 90, 72, 60, \dots$.

Step 2: Complete Sequences

Summarizing the sequences:

1. Number of sides: $3, 4, 5, 6, \dots$
2. Sum of interior angles: $180, 360, 540, 720, \dots$
3. Sum of exterior angles: $360, 360, 360, 360, \dots$ (constant)
4. One interior angle: $60, 90, 108, 120, \dots$
5. One exterior angle: $120, 90, 72, 60, \dots$

Would you like further clarifications or additional examples?

Relative Questions

1. How is the sum of interior angles related to the number of sides of a polygon?
2. Why is the sum of exterior angles constant for all polygons?
3. What happens to the interior and exterior angles as the number of sides increases?
4. Can you derive these sequences for a heptagon ($n=7$) or octagon ($n=8$)?
5. How do these concepts apply to irregular polygons?

Tip

Always remember that regular polygons have equal-length sides and angles, which simplifies calculations for their properties!