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The sequence in the image is \( 3, 12, 21, 30, \dots \). This is an arithmetic sequence where each term increases by the same value. Let's break down how to find the rule for the \(n\)-th term.

1. **Find the common difference**:
   - The difference between consecutive terms is \( 12 - 3 = 9 \), \( 21 - 12 = 9 \), and \( 30 - 21 = 9 \). So, the common difference \(d = 9\).

2. **Find the first term**:
   - The first term of the sequence is \( a_1 = 3 \).

3. **Use the formula for the \(n\)-th term of an arithmetic sequence**:
   \[
   a_n = a_1 + (n - 1) \cdot d
   \]
   Substituting \(a_1 = 3\) and \(d = 9\):
   \[
   a_n = 3 + (n - 1) \cdot 9
   \]
   Simplifying:
   \[
   a_n = 3 + 9n - 9 = 9n - 6
   \]

Thus, the \(n\)-th term rule for this sequence is \( 9n - 6 \).

Would you like more details or clarifications on any of these steps?

Here are 5 related questions to expand your understanding:
1. What is the general formula for the \(n\)-th term of an arithmetic sequence?
2. How would the rule change if the first term was different?
3. Can you find the 10th term using the formula \( 9n - 6 \)?
4. What if the common difference was negative? How would the sequence behave?
5. How do you determine if a sequence is arithmetic or geometric?

**Tip**: Always check the first few differences between terms to identify if a sequence is arithmetic (constant difference) or geometric (constant ratio).

The start of an arithmetic sequence is shown below. What is the nth term rule for this sequence?

Learn how to find the nth term rule for an arithmetic sequence starting with the terms 3, 12, 21, and 30. This problem covers key concepts in arithmetic sequences, including how to calculate the common difference and apply the nth term formula. Suitable for students in Grades 6-8.

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