The problem presents a sequence defined by $u_0 = 12$ and the recurrence relation:

$u_{n+1} = \frac{1}{3}u_n + 2$

Question 1: Using the graph, conjecture the behavior of the sequence $(u_n)$.

The graph shows two lines:

1. Blue line: $y = x$, which is the identity line. It can represent a fixed value where $u_{n+1} = u_n$.
2. Red line: $y = \frac{1}{3}x + 2$, which represents the recurrence relation.

Observing the graph, you can see that the sequence starts at $u_0 = 12$ and the next values of $u_n$ are generated by applying the recurrence relation. Since the slope of the line $y = \frac{1}{3}x + 2$ is less than 1, the sequence will decrease over time but approach a limiting value (asymptote).

Conjecture:

The sequence $(u_n)$ is decreasing and will eventually stabilize towards a limit, which can be guessed from the intersection of the lines. The limit is found when $u_{n+1} = u_n$, so:

$u = \frac{1}{3}u + 2$

Solving for $u$:

$u - \frac{1}{3}u = 2 \implies \frac{2}{3}u = 2 \implies u = 3$

Thus, the sequence tends towards the value 3.

Question 2: Prove the conjecture by induction.

We will prove that the sequence is decreasing, i.e., $u_{n+1} \leq u_n$, and show that it converges to the value 3.

Base Case:

For $n = 0$:

$u_1 = \frac{1}{3}u_0 + 2 = \frac{1}{3} \times 12 + 2 = 4 + 2 = 6$

Clearly, $u_1 = 6 \leq 12 = u_0$.

Induction Hypothesis:

Assume that $u_n \leq u_{n-1}$ for some $n \geq 1$.

Induction Step:

We need to show that $u_{n+1} \leq u_n$. From the recurrence relation:

$u_{n+1} = \frac{1}{3}u_n + 2$

We know that $u_n \geq 3$, since the sequence is decreasing and approaches 3. We need to check that $u_{n+1} \leq u_n$:

$u_{n+1} = \frac{1}{3}u_n + 2 \leq u_n$

This simplifies to:

$\frac{1}{3}u_n + 2 \leq u_n$

Subtracting $\frac{1}{3}u_n$ from both sides:

$2 \leq \frac{2}{3}u_n$

Multiplying both sides by $\frac{3}{2}$:

$3 \leq u_n$

Thus, the sequence is decreasing and bounded below by 3. Therefore, by the principle of mathematical induction, the sequence converges to 3.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What happens to the sequence if the initial value $u_0$ is changed?
2. How would the recurrence relation change if the constant 2 in $u_{n+1} = \frac{1}{3}u_n + 2$ were different?
3. How can you determine if a recurrence relation has a limit without graphing it?
4. Can you express the general term $u_n$ explicitly without using recurrence?
5. What other types of sequences have similar long-term behavior (e.g., geometric or arithmetic)?

Tip: When working with recurrence relations, try solving for the fixed point (limit) by setting $u_{n+1} = u_n$.