https://file.aiquickdraw.com/mathbot/file/1725974207148kt3dmd3j.jpg

The image you have shared represents three types of polygonal numbers: **Square Numbers**, **Pentagonal Numbers**, and **Hexagonal Numbers**. Let’s break down how each type of polygonal number is formed and how to solve problems involving these numbers.

### 1. **Square Numbers**:
- These numbers are formed by arranging dots to form squares. The formula to calculate the \(n\)-th square number is:
  \[
  S_n = n^2
  \]
- Example from the image: The first few square numbers are:
  - \(1^2 = 1\)
  - \(2^2 = 4\)
  - \(3^2 = 9\)
  - \(4^2 = 16\)

### 2. **Pentagonal Numbers**:
- Pentagonal numbers are represented by dots arranged in a pentagon. The formula to find the \(n\)-th pentagonal number is:
  \[
  P_n = \frac{3n^2 - n}{2}
  \]
- Example from the image: The first few pentagonal numbers are:
  - \(P_1 = 1\)
  - \(P_2 = 5\)
  - \(P_3 = 12\)
  - \(P_4 = 22\)
  - \(P_5 = 35\)

### 3. **Hexagonal Numbers**:
- Hexagonal numbers are formed by arranging dots in the shape of a hexagon. The formula for the \(n\)-th hexagonal number is:
  \[
  H_n = 2n^2 - n
  \]
- Example from the image: The first few hexagonal numbers are:
  - \(H_1 = 1\)
  - \(H_2 = 6\)
  - \(H_3 = 15\)
  - \(H_4 = 28\)
  - \(H_5 = 45\)

### General Solution Strategy:
- **Identify the number type**: Use the image to determine if the sequence represents square, pentagonal, or hexagonal numbers.
- **Apply the corresponding formula**: Based on the number type (square, pentagonal, or hexagonal), use the appropriate formula to compute the required values.

Would you like more detailed examples of how to use these formulas, or do you have specific questions related to these numbers?

### 5 Relative Questions:
1. How can you find the 10th pentagonal number using the formula?
2. What is the difference between the 5th hexagonal number and the 5th square number?
3. How can you generalize the relationship between the sum of square numbers?
4. Can hexagonal numbers be represented in a different geometric form?
5. What are the real-life applications of pentagonal and hexagonal numbers?

### Tip:
Polygonal numbers are useful for recognizing patterns and solving problems in number theory and combinatorics.

Learn how to solve problems using polygonal numbers such as square, pentagonal, and hexagonal numbers. Explore the mathematical formulas for each type and how they are visually represented. Suitable for Grades 6-9.

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