Let's analyze and solve the problem.

We are given the recurrence relation for a sequence $u_{n+2} = u_{n+1} + u_n$, with initial conditions $u_1 = 1$ and $u_2 = 1$.

Part (a): Listing the first eight terms of the sequence

Using the recurrence relation, we can compute each term as follows:

• $u_1 = 1$
• $u_2 = 1$
• $u_3 = u_2 + u_1 = 1 + 1 = 2$
• $u_4 = u_3 + u_2 = 2 + 1 = 3$
• $u_5 = u_4 + u_3 = 3 + 2 = 5$
• $u_6 = u_5 + u_4 = 5 + 3 = 8$
• $u_7 = u_6 + u_5 = 8 + 5 = 13$
• $u_8 = u_7 + u_6 = 13 + 8 = 21$

Thus, the first eight terms are:

$u_1 = 1, \ u_2 = 1, \ u_3 = 2, \ u_4 = 3, \ u_5 = 5, \ u_6 = 8, \ u_7 = 13, \ u_8 = 21.$

Part (b): Name of the sequence

This sequence is known as the Fibonacci sequence.

Part (c): Position of the number 233 in the sequence

Continuing the sequence from $u_9$ onward:

• $u_9 = u_8 + u_7 = 21 + 13 = 34$
• $u_{10} = u_9 + u_8 = 34 + 21 = 55$
• $u_{11} = u_{10} + u_9 = 55 + 34 = 89$
• $u_{12} = u_{11} + u_{10} = 89 + 55 = 144$
• $u_{13} = u_{12} + u_{11} = 144 + 89 = 233$

Thus, the number 233 appears at position 13 in the sequence.

Do you have any questions or need further clarification?

Here are 5 related questions to explore:

1. What is the closed-form formula for the Fibonacci sequence?
2. How can you generalize the recurrence relation to different initial conditions?
3. What is the relationship between Fibonacci numbers and the golden ratio?
4. Can you derive a formula to calculate the $n$-th Fibonacci number directly?
5. How can Fibonacci numbers be found in nature and art?

Tip: The Fibonacci sequence is often used to model growth patterns, such as in population studies or natural phenomena like spirals in shells and flowers.