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The image you've provided showcases three types of polygonal numbers: square numbers, pentagonal numbers, and hexagonal numbers. I'll explain each type of polygonal number and how to solve for them using formulas.

### 1. **Square Numbers**
Square numbers represent dots arranged in a square pattern, and they follow the formula:
\[
S_n = n^2
\]
Where:
- \(S_n\) is the \(n\)-th square number.
- \(n\) is the term position.

For example:
- For \(n = 1\): \(S_1 = 1^2 = 1\)
- For \(n = 2\): \(S_2 = 2^2 = 4\)
- For \(n = 3\): \(S_3 = 3^2 = 9\)
- For \(n = 4\): \(S_4 = 4^2 = 16\)

### 2. **Pentagonal Numbers**
Pentagonal numbers are numbers that can be represented as dots forming a pentagon. The formula for the \(n\)-th pentagonal number is:
\[
P_n = \frac{3n^2 - n}{2}
\]
Where:
- \(P_n\) is the \(n\)-th pentagonal number.

For example:
- For \(n = 1\): \(P_1 = \frac{3(1)^2 - 1}{2} = 1\)
- For \(n = 2\): \(P_2 = \frac{3(2)^2 - 2}{2} = 5\)
- For \(n = 3\): \(P_3 = \frac{3(3)^2 - 3}{2} = 12\)
- For \(n = 4\): \(P_4 = \frac{3(4)^2 - 4}{2} = 22\)
- For \(n = 5\): \(P_5 = \frac{3(5)^2 - 5}{2} = 35\)

### 3. **Hexagonal Numbers**
Hexagonal numbers are numbers that can be represented as dots forming a hexagon. The formula for the \(n\)-th hexagonal number is:
\[
H_n = 2n^2 - n
\]
Where:
- \(H_n\) is the \(n\)-th hexagonal number.

For example:
- For \(n = 1\): \(H_1 = 2(1)^2 - 1 = 1\)
- For \(n = 2\): \(H_2 = 2(2)^2 - 2 = 6\)
- For \(n = 3\): \(H_3 = 2(3)^2 - 3 = 15\)
- For \(n = 4\): \(H_4 = 2(4)^2 - 4 = 28\)
- For \(n = 5\): \(H_5 = 2(5)^2 - 5 = 45\)

### Summary of the given sequences:
- **Square Numbers**: 1, 4, 9, 16, ...
- **Pentagonal Numbers**: 1, 5, 12, 22, 35, ...
- **Hexagonal Numbers**: 1, 6, 15, 28, 45, ...

Would you like to see detailed calculations or have any specific questions about these sequences?

Here are some related questions:
1. What is the geometric representation of the 6th pentagonal number?
2. How do square numbers relate to area in geometry?
3. Can you derive the formula for polygonal numbers in general?
4. What is the 7th hexagonal number, and how is it calculated?
5. How are triangular numbers connected to these polygonal numbers?

**Tip**: All polygonal numbers, including square, pentagonal, and hexagonal, can be visualized geometrically, helping to better understand their patterns and relationships.

This page provides a detailed explanation of polygonal numbers, focusing on square, pentagonal, and hexagonal numbers. Learn the formulas for calculating these numbers and explore their geometric representations. Suitable for students in Grades 7-9.

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