https://file.aiquickdraw.com/mathbot/file/1726510734148rv80scd1.jpg

Let's solve the two sequences provided:

### First Sequence: \( 9, 16, 23, 30, 37, \ldots \)

1. **Pattern Analysis:**
   - The first term is \( 9 \), and the common difference between the terms is \( 7 \) (since \( 16 - 9 = 7 \), \( 23 - 16 = 7 \), etc.).

2. **General Formula:**
   - This sequence is arithmetic, with a first term \( a = 9 \) and a common difference \( d = 7 \).
   - The general formula for the \( n \)-th term of an arithmetic sequence is:
     \[
     a_n = a + (n - 1) \cdot d
     \]
   - Substituting the values of \( a = 9 \) and \( d = 7 \):
     \[
     a_n = 9 + (n - 1) \cdot 7 = 9 + 7n - 7 = 7n + 2
     \]
   - Therefore, the formula for the \( n \)-th term of this sequence is:
     \[
     a_n = 7n + 2
     \]

---

### Second Sequence: \( 2, 0, -2, -4, -6, \ldots \)

1. **Pattern Analysis:**
   - The first term is \( 2 \), and the common difference between the terms is \( -2 \) (since \( 0 - 2 = -2 \), \( -2 - 0 = -2 \), etc.).

2. **General Formula:**
   - This sequence is also arithmetic, with a first term \( a = 2 \) and a common difference \( d = -2 \).
   - Using the general arithmetic formula:
     \[
     a_n = a + (n - 1) \cdot d
     \]
   - Substituting the values of \( a = 2 \) and \( d = -2 \):
     \[
     a_n = 2 + (n - 1) \cdot (-2) = 2 - 2n + 2 = 4 - 2n
     \]
   - Therefore, the formula for the \( n \)-th term of this sequence is:
     \[
     a_n = 4 - 2n
     \]

---

### Final Results:
1. For the first sequence: \( a_n = 7n + 2 \)
2. For the second sequence: \( a_n = 4 - 2n \)

Would you like more details on how these formulas were derived? 

Here are some related questions to think about:
1. What is the 10th term of each sequence?
2. How would the formula change if the first term was different but the common difference stayed the same?
3. Can you derive the general formula for an arithmetic sequence with any first term \( a \) and common difference \( d \)?
4. How can you identify if a sequence is arithmetic or geometric?
5. What are the applications of arithmetic sequences in real life?

**Tip:** Always look for common differences or ratios when analyzing number patterns; they help in identifying the sequence type quickly!

Write down a formula for the nth term of these patterns: (1) 9, 16, 23, 30, 37, ... (2) 2, 0, -2, -4, -6, ...

Learn how to derive the nth term formulas for two arithmetic sequences: (1) 9, 16, 23... and (2) 2, 0, -2... This solution covers arithmetic sequence concepts and general formula derivation. Suitable for Grades 7-9.

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