To determine the values of the terms \( u_4 = -4 \) and \( u_{15} = 29 \), we can analyze this as an arithmetic sequence, where the difference between consecutive terms is constant. Here are the steps:

1. **Identify Known Terms:**
   - We know \( u_4 = -4 \).
   - We know \( u_{15} = 29 \).

2. **General Form of an Arithmetic Sequence:**
   - In an arithmetic sequence, each term can be represented as:
     \[
     u_n = u_1 + (n - 1)d
     \]
     where:
     - \( u_1 \) is the first term,
     - \( d \) is the common difference,
     - \( n \) is the term number.

3. **Setting Up Equations:**
   - For \( u_4 \):
     \[
     u_4 = u_1 + (4 - 1)d = -4
     \]
   - For \( u_{15} \):
     \[
     u_{15} = u_1 + (15 - 1)d = 29
     \]

4. **Solving the System of Equations:**
   - Expanding both equations:
     \[
     u_1 + 3d = -4
     \]
     \[
     u_1 + 14d = 29
     \]
   - Subtract the first equation from the second to eliminate \( u_1 \):
     \[
     (u_1 + 14d) - (u_1 + 3d) = 29 - (-4)
     \]
     \[
     11d = 33
     \]
     \[
     d = 3
     \]

5. **Finding \( u_1 \):**
   - Substitute \( d = 3 \) into the first equation:
     \[
     u_1 + 3(3) = -4
     \]
     \[
     u_1 + 9 = -4
     \]
     \[
     u_1 = -13
     \]

6. **Summary of Results:**
   - The first term \( u_1 = -13 \).
   - The common difference \( d = 3 \).

The sequence can now be defined by:
\[
u_n = -13 + (n - 1) \cdot 3
\]

Would you like further details on this process or to verify specific terms in the sequence?

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Here are five questions you might explore next:
1. What is the 10th term of this sequence?
2. How many terms are needed for the sequence to reach a value above 50?
3. What is the sum of the first 15 terms?
4. If the sequence continues, what would be the 100th term?
5. Can this sequence be graphed, and what would its pattern look like?

**Tip:** In arithmetic sequences, each term increases (or decreases) by a constant value, making them easy to analyze with simple formulas.

Learn to solve an arithmetic sequence problem where u4 = -4 and u15 = 29, finding the first term and common difference. This solution includes step-by-step explanations of setting up equations and solving for unknowns. Suitable for students in Grades 8-10.

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