Write a Pattern rule that uses algebra to describe each pattern

To write a pattern rule using algebra, we can observe how the numbers in each sequence change from one term to the next. In general, we look for the relationship between the term position (often represented by \( n \)) and the value of the term.

Since I can't directly see the details of your image, I'll provide a step-by-step method to create an algebraic rule for any number pattern:

### Steps to Find an Algebraic Pattern Rule:
1. **Identify the Pattern Type**: Check if the sequence is arithmetic (adding/subtracting a fixed number) or geometric (multiplying/dividing by a fixed number). If neither, look for other relationships.

2. **Define the Term Position**: Assign each term a position \( n \) (starting with 1, 2, 3, ...).

3. **Find the Difference or Ratio**: If it's an arithmetic sequence, find the difference between terms (common difference \( d \)). If it's geometric, find the ratio.

4. **Create a Formula**:
   - **For an Arithmetic Sequence**: Use \( a_n = a_1 + (n - 1) \times d \), where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, and \( d \) is the common difference.
   - **For a Geometric Sequence**: Use \( a_n = a_1 \times r^{(n - 1)} \), where \( r \) is the common ratio.

5. **Test and Refine the Formula**: Substitute values to ensure the formula matches all terms.

### Example:
If your sequence is **2, 5, 8, 11,...**
1. **Pattern Type**: Arithmetic (each term increases by 3).
2. **Difference**: Common difference \( d = 3 \).
3. **Formula**: Since the first term \( a_1 = 2 \), use \( a_n = 2 + (n - 1) \times 3 \).
4. **Simplify**: \( a_n = 3n - 1 \).

So, the algebraic rule for this pattern is **\( a_n = 3n - 1 \)**.

If you'd like, upload the image with the pattern details, and I can help create the specific rule based on that information. Let me know if you want further details or examples!

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### Additional Questions for Practice:
1. What is the algebraic rule for a sequence with terms 4, 7, 10, 13,...?
2. How do you derive an algebraic rule for a geometric sequence with terms 3, 6, 12,...?
3. How can you modify an algebraic pattern rule if the sequence starts from a different number?
4. How would you write the rule for an alternating pattern (like 2, -2, 2, -2,...)?
5. What is the algebraic formula for a sequence defined by a quadratic pattern?

#### Tip:
For patterns that don’t fit linear or geometric types, consider if they follow a quadratic form, where the rule might involve \( n^2 \) or higher powers.

Learn how to write an algebraic rule for arithmetic and geometric sequences. This step-by-step guide covers key algebraic formulas for identifying patterns and finding general rules. Suitable for students in Grades 6-8.

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