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The image you uploaded displays three different types of polygonal numbers: square numbers, pentagonal numbers, and hexagonal numbers. These are examples of figurate numbers, which are numbers that can be arranged in regular polygonal shapes. I'll explain how each set of numbers is calculated and how you can identify them using formulas.

### 1. **Square Numbers**:
Square numbers are numbers that can form a perfect square. The pattern follows a \( n \times n \) arrangement.

- **Formula**: 
  \[
  S_n = n^2
  \]
  where \( S_n \) is the \( n \)-th square number, and \( n \) is a positive integer.
  
  For example:
  - \( S_1 = 1^2 = 1 \)
  - \( S_2 = 2^2 = 4 \)
  - \( S_3 = 3^2 = 9 \)
  - \( S_4 = 4^2 = 16 \)

  The numbers shown in the diagram represent squares of size 1x1, 2x2, 3x3, etc.

### 2. **Pentagonal Numbers**:
Pentagonal numbers are numbers that can be arranged in the shape of a pentagon. These numbers follow a specific growth pattern.

- **Formula**:
  \[
  P_n = \frac{3n^2 - n}{2}
  \]
  where \( P_n \) is the \( n \)-th pentagonal number.
  
  For example:
  - \( P_1 = \frac{3(1)^2 - 1}{2} = 1 \)
  - \( P_2 = \frac{3(2)^2 - 2}{2} = 5 \)
  - \( P_3 = \frac{3(3)^2 - 3}{2} = 12 \)
  - \( P_4 = \frac{3(4)^2 - 4}{2} = 22 \)
  - \( P_5 = \frac{3(5)^2 - 5}{2} = 35 \)

### 3. **Hexagonal Numbers**:
Hexagonal numbers form a hexagon shape and follow a different growth pattern from pentagonal numbers.

- **Formula**:
  \[
  H_n = 2n^2 - n
  \]
  where \( H_n \) is the \( n \)-th hexagonal number.
  
  For example:
  - \( H_1 = 2(1)^2 - 1 = 1 \)
  - \( H_2 = 2(2)^2 - 2 = 6 \)
  - \( H_3 = 2(3)^2 - 3 = 15 \)
  - \( H_4 = 2(4)^2 - 4 = 28 \)
  - \( H_5 = 2(5)^2 - 5 = 45 \)

### Summary:
Each of these polygonal numbers has a specific formula based on its shape:
- Square numbers follow \( n^2 \),
- Pentagonal numbers follow \( \frac{3n^2 - n}{2} \),
- Hexagonal numbers follow \( 2n^2 - n \).

These formulas allow you to calculate the numbers for any \( n \) and understand how they form geometric patterns.

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Would you like further details on any of these shapes or formulas?

Here are five follow-up questions to deepen your understanding:
1. How do polygonal numbers relate to triangular numbers?
2. What is the relationship between pentagonal numbers and the sums of consecutive integers?
3. Can you derive a formula for a heptagonal number (7-sided polygon)?
4. How are these numbers used in number theory, particularly for solving Diophantine equations?
5. Can any polygonal number be a perfect square?

**Tip**: Visualizing figurate numbers can help you understand the underlying patterns and growth of these number sequences more clearly. Try drawing them out to see the patterns form geometrically!

Learn how to solve problems using polygonal numbers, including square, pentagonal, and hexagonal numbers. This guide covers the formulas, geometric patterns, and theorems needed to understand polygonal number sequences. Suitable for students in Grades 8-10.

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