The given recurrence relation is:

\[
U_{n+1} = \sqrt{2 + U_n}
\]

This describes a sequence where each term depends on the square root of 2 plus the previous term. To analyze or solve this, we can approach it in several ways:

### 1. **Finding the Fixed Point:**
We can find a fixed point by solving for \(U_n = U_{n+1}\), meaning the sequence reaches a stable value.

\[
U = \sqrt{2 + U}
\]

Squaring both sides to eliminate the square root:

\[
U^2 = 2 + U
\]

Rearranging this into a quadratic equation:

\[
U^2 - U - 2 = 0
\]

Factoring:

\[
(U - 2)(U + 1) = 0
\]

Thus, the possible fixed points are:

\[
U = 2 \quad \text{or} \quad U = -1
\]

### 2. **Behavior of the Sequence:**
- If the initial value \(U_0 \geq 2\), the sequence tends to stabilize at \( U = 2 \).
- If \( U_0 \) is less than -1, the recurrence relation would produce complex numbers due to the square root of a negative value.
- If \( -1 < U_0 < 2 \), the sequence still tends to \(U = 2\) because the square root term will "pull" the sequence upward over time.

### 3. **First Few Terms of the Sequence:**
To get a sense of how the sequence evolves, you can calculate the first few terms for a given initial value \(U_0\).

For example, if \(U_0 = 1\):
\[
U_1 = \sqrt{2 + 1} = \sqrt{3} \approx 1.732
\]
\[
U_2 = \sqrt{2 + 1.732} = \sqrt{3.732} \approx 1.931
\]

The values slowly approach 2.

Would you like to see more terms for a specific initial value, or a deeper analysis?

### Relative Questions:
1. How would the sequence behave if \(U_0 = 0\)?
2. What happens if the initial value is negative?
3. How can we verify if the sequence converges for any initial value \(U_0\)?
4. How does the rate of convergence depend on the initial value?
5. Can this recurrence relation be represented in a closed form for large \(n\)?

### Tip:
Always check for fixed points in recurrence relations to understand the long-term behavior of the sequence.

Analyze the recurrence relation Un+1 = (2 + Un)^0.5 and discover its fixed points and convergence behavior. Learn how to solve quadratic equations and understand how initial values influence sequence behavior. Suitable for high school students in Grades 10-12.

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