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?

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.