https://file.aiquickdraw.com/mathbot/file/17285734678503wse234p.jpg

Let's solve both arithmetic sequence problems given in the image:

### Problem 15
The sequence provided is \( \{5, 3, 1, -1, \dots\} \).

1. **Identify the first term (\(a_1\))**: \(a_1 = 5\).
2. **Find the common difference (\(d\))**: 
   \[
   d = 3 - 5 = -2
   \]
3. **Use the formula for the \(n\)-th term of an arithmetic sequence**: 
   \[
   a_n = a_1 + (n - 1)d
   \]
4. **Substitute values to find \(a_{18}\)**:
   \[
   a_{18} = 5 + (18 - 1)(-2)
   \]
   \[
   a_{18} = 5 + 17(-2)
   \]
   \[
   a_{18} = 5 - 34
   \]
   \[
   a_{18} = -29
   \]
   **Answer**: \( a_{18} = -29 \)

### Problem 16
The sequence provided is \( \{-2, 5, 12, 19, \dots\} \).

1. **Identify the first term (\(a_1\))**: \(a_1 = -2\).
2. **Find the common difference (\(d\))**:
   \[
   d = 5 - (-2) = 7
   \]
3. **Use the formula for the \(n\)-th term of an arithmetic sequence**:
   \[
   a_n = a_1 + (n - 1)d
   \]
4. **Substitute values to find \(a_{34}\)**:
   \[
   a_{34} = -2 + (34 - 1)(7)
   \]
   \[
   a_{34} = -2 + 33 \times 7
   \]
   \[
   a_{34} = -2 + 231
   \]
   \[
   a_{34} = 229
   \]
   **Answer**: \( a_{34} = 229 \)

Let me know if you need further clarification or additional details!

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#### Related Questions:
1. How do you find the sum of the first \( n \) terms in an arithmetic sequence?
2. What are the conditions under which a sequence is considered arithmetic?
3. How do you derive the formula for the \(n\)-th term of an arithmetic sequence?
4. Can an arithmetic sequence have a common difference of zero?
5. How do you solve for \(n\) in an arithmetic sequence if you know the value of \(a_n\)?

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#### Tip:
In arithmetic sequences, if you know two terms and their positions, you can always determine the common difference and use it to find other terms.

Given {5, 3, 1, -1, ...}, Find a18 and Given {-2, 5, 12, 19, ...}, Find a34.

Learn how to find the 18th and 34th terms in arithmetic sequences. Detailed step-by-step solutions using the formula a_n = a_1 + (n - 1)d, ideal for Grades 8-10.

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