The given recurrence relation is:

$U_1 = 1$ $U_{n+1} = \frac{U_n}{1 + n U_n} \quad \text{for} \quad n = 1, 2, 3, \dots$

You want to find $U_{2025}$ using this recurrence. We will first calculate the first few terms to try and understand the pattern and behavior of the sequence.

Step 1: Calculate $U_2$

For $n = 1$: $U_2 = \frac{U_1}{1 + 1 \cdot U_1} = \frac{1}{1 + 1 \cdot 1} = \frac{1}{2}$

Step 2: Calculate $U_3$

For $n = 2$: $U_3 = \frac{U_2}{1 + 2 \cdot U_2} = \frac{\frac{1}{2}}{1 + 2 \cdot \frac{1}{2}} = \frac{\frac{1}{2}}{1 + 1} = \frac{\frac{1}{2}}{2} = \frac{1}{4}$

Step 3: Calculate $U_4$

For $n = 3$: $U_4 = \frac{U_3}{1 + 3 \cdot U_3} = \frac{\frac{1}{4}}{1 + 3 \cdot \frac{1}{4}} = \frac{\frac{1}{4}}{1 + \frac{3}{4}} = \frac{\frac{1}{4}}{\frac{7}{4}} = \frac{1}{7}$

Step 4: Calculate $U_5$

For $n = 4$: $U_5 = \frac{U_4}{1 + 4 \cdot U_4} = \frac{\frac{1}{7}}{1 + 4 \cdot \frac{1}{7}} = \frac{\frac{1}{7}}{1 + \frac{4}{7}} = \frac{\frac{1}{7}}{\frac{11}{7}} = \frac{1}{11}$

Observing the Pattern:

From the above steps, we can see that the sequence follows the pattern: $U_1 = 1, \quad U_2 = \frac{1}{2}, \quad U_3 = \frac{1}{4}, \quad U_4 = \frac{1}{7}, \quad U_5 = \frac{1}{11}, \dots$ It appears that the denominator is increasing in a pattern of the sequence of triangular numbers: $1, 2, 3, 4, 5, \dots$.

The general form for $U_n$ is: $U_n = \frac{1}{T_n} \quad \text{where} \quad T_n \text{ is the n-th triangular number}.$

The $n$-th triangular number $T_n$ is given by: $T_n = \frac{n(n+1)}{2}$

Step 5: Find $U_{2025}$

For $n = 2025$, the triangular number $T_{2025}$ is: $T_{2025} = \frac{2025(2025 + 1)}{2} = \frac{2025 \cdot 2026}{2} = 2049393$

Thus: $U_{2025} = \frac{1}{T_{2025}} = \frac{1}{2049393}$

Final Answer:

$U_{2025} = \frac{1}{2049393}$

Would you like further clarification on any part of this process, or do you have additional questions?

Additional Questions:

1. How do triangular numbers relate to other types of sequences in mathematics?
2. Can this recurrence be generalized to other forms of relations?
3. What is the behavior of this sequence for large $n$?
4. How do we calculate triangular numbers efficiently?
5. Can you derive a closed-form expression for this sequence without computing each term?

Tip: When dealing with recurrence relations, always try to identify patterns in the terms to make solving easier, especially when the recurrence involves ratios or simple algebraic expressions.