Let's solve the problem step by step using the given recursive formula.

### Given:
- **U1 = 1**
- **Un = Un-1 + floor(Un-1 / 4)** for n ≥ 2

Where **floor(x)** denotes the greatest integer less than or equal to **x**.

### Step 1: Find **U2**
\[
U2 = U1 + \text{floor}(U1 / 4)
\]
\[
U2 = 1 + \text{floor}(1 / 4) = 1 + 0 = 1
\]

### Step 2: Find **U3**
\[
U3 = U2 + \text{floor}(U2 / 4)
\]
\[
U3 = 1 + \text{floor}(1 / 4) = 1 + 0 = 1
\]

### Step 3: Find **U4**
\[
U4 = U3 + \text{floor}(U3 / 4)
\]
\[
U4 = 1 + \text{floor}(1 / 4) = 1 + 0 = 1
\]

### Conclusion:
The values are:
- **U2 = 1**
- **U3 = 1**
- **U4 = 1**

The sequence remains constant at **1** for the first four terms.

Would you like further details or have any questions?

---

**Follow-up Questions:**
1. Can you predict what **U5** and **U6** will be based on this pattern?
2. How does the greatest integer function (floor) influence this sequence?
3. Can this sequence ever increase? Why or why not?
4. What changes would you expect if the initial term **U1** was different?
5. How would the sequence differ if the division was by a number other than four?

**Tip:** When analyzing sequences, observe whether a pattern stabilizes and how recursive definitions impact the sequence's growth or stability.

Explore how to solve the recursive sequence problem U(n) = U(n-1) + floor(U(n-1) / 4) step by step. Understand the impact of the floor function on sequence values and analyze the stability of the sequence. Suitable for high school students studying recurrence relations and mathematical sequences.

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