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The sequence \( \{u_n\} \) is defined as follows:

\[
u_0 = 1, \quad u_1 = 2, \quad u_{n+1} = 3u_n - 2u_{n-1} \quad \text{for} \quad n \in \mathbb{Z}^+.
\]

We are tasked with finding \( u_2 \), \( u_3 \), and \( u_4 \).

### Step-by-step Solution:

1. **For \( u_2 \):**
   Using the recurrence relation:
   \[
   u_2 = 3u_1 - 2u_0
   \]
   Substitute the known values \( u_1 = 2 \) and \( u_0 = 1 \):
   \[
   u_2 = 3(2) - 2(1) = 6 - 2 = 4.
   \]

2. **For \( u_3 \):**
   Using the recurrence relation:
   \[
   u_3 = 3u_2 - 2u_1
   \]
   Substitute the known values \( u_2 = 4 \) and \( u_1 = 2 \):
   \[
   u_3 = 3(4) - 2(2) = 12 - 4 = 8.
   \]

3. **For \( u_4 \):**
   Using the recurrence relation:
   \[
   u_4 = 3u_3 - 2u_2
   \]
   Substitute the known values \( u_3 = 8 \) and \( u_2 = 4 \):
   \[
   u_4 = 3(8) - 2(4) = 24 - 8 = 16.
   \]

### Final Answers:
- \( u_2 = 4 \)
- \( u_3 = 8 \)
- \( u_4 = 16 \)

Would you like further details or have any questions?

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Here are 5 related questions to expand on this problem:
1. Can you find the general term \( u_n \) for this sequence?
2. How would you approach solving this sequence using matrix methods?
3. What is the significance of the recurrence relation in sequences?
4. How does the behavior of the sequence change if we modify the initial conditions \( u_0 \) and \( u_1 \)?
5. Can you express the sequence as a summation formula?

**Tip:** For recurrence relations, always start by substituting known values to build the next terms in the sequence step by step.

A sequence {u_n} is defined by u_0 = 1, u_1 = 2, u_{n+1} = 3u_n - 2u_{n-1} for n ∈ Z+. Find u_2, u_3, and u_4.

Learn how to find terms in a recurrence sequence defined by u_0 = 1, u_1 = 2, and u_{n+1} = 3u_n - 2u_{n-1}. Detailed steps for solving u_2, u_3, and u_4, suitable for high school algebra students.

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