https://file.aiquickdraw.com/mathbot/file/1727316734992jupbj23x.jpg

The image shows a pattern of alternating black and white squares in three steps, and the question asks for explicit equations that represent the pattern for:

- **Part (a)**: Black and white squares in each step.
- **Part (b)**: Pink dots (though this part is not visible in the provided portion).

### Step-by-step analysis of **part (a)**:

1. **Step 1**: 
   - Sequence: ⚫◻️◼️◻️ (4 squares)
2. **Step 2**: 
   - Sequence: ⚫◻️◼️◻️⚫◻️ (6 squares)
3. **Step 3**: 
   - Sequence: ⚫◻️◼️◻️⚫◻️◼️◻️ (8 squares)

### Observing the pattern:
- At each step, the number of squares (both black and white) increases by 2.
- The number of squares in Step \(n\) seems to be given by: 
  \[
  \text{Number of squares} = 2n + 2
  \]
  
For **Step 1**, \(n = 1\) gives 4 squares. For **Step 2**, \(n = 2\) gives 6 squares, and for **Step 3**, \(n = 3\) gives 8 squares, which matches the given patterns.

### Explicit equation for **part (a)**:
For the black and white square pattern, the explicit equation that represents the total number of squares in Step \(n\) is:
\[
\text{Total squares in Step } n = 2n + 2
\]

### Would you like me to explain or confirm the equation further?

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Here are 5 follow-up questions that may help expand this concept:

1. How would the equation change if the squares increased by 3 at each step?
2. Can you find the number of black squares specifically at each step?
3. How does the sequence of white squares relate to the total number of squares?
4. If the pattern extended to Step 10, how many total squares would there be?
5. How would you modify the equation if the initial pattern started with 6 squares instead?

**Tip**: Identifying how the pattern grows (additively, multiplicatively, etc.) is key to deriving equations. Always check the consistency of the equation with several steps in the pattern!

Write an explicit equation that represents the pattern of black and white squares in the given sequence (Steps 1-3).

Learn how to derive an explicit equation for a pattern involving black and white squares, with a solution based on arithmetic sequences. This math problem demonstrates how the number of squares increases step by step. Suitable for students in Grades 6-8.

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