The given recursive relation is:

\[
U_{n+1} = (U_n + 10)(U_n + 8)
\]

This equation describes how each subsequent term \( U_{n+1} \) in the sequence is generated from the previous term \( U_n \).

### Steps to analyze the recursive relation:
1. **Expression Structure**:
   \[
   U_{n+1} = (U_n + 10)(U_n + 8)
   \]
   - This is a product of two binomials, which can be expanded:
   \[
   U_{n+1} = U_n^2 + 18U_n + 80
   \]

   This form tells us that each term in the sequence is determined by squaring the previous term, adding a linear component (\(18U_n\)), and a constant (80).

2. **Type of Sequence**:
   - This recursive relation defines a **quadratic recurrence**, meaning the next term depends on the square of the current term, along with linear and constant adjustments.
   - The nature of this sequence will heavily depend on the initial condition, \( U_0 \), or the first term.

3. **Behavior of the Sequence**:
   - For certain initial values \( U_0 \), the terms in the sequence could grow very rapidly since squaring a term tends to make the sequence grow quickly.
   - Depending on \( U_0 \), the sequence could either increase indefinitely, remain bounded, or behave in more complex ways.

4. **Finding Terms of the Sequence**:
   - To compute specific terms, you'd need the initial value \( U_0 \). From there, you can compute \( U_1 \), \( U_2 \), and so on.

   For example:
   If \( U_0 = 0 \),
   \[
   U_1 = (0 + 10)(0 + 8) = 10 \times 8 = 80
   \]
   Then,
   \[
   U_2 = (80 + 10)(80 + 8) = 90 \times 88 = 7920
   \]

Would you like to explore specific examples with a given initial condition, or analyze the long-term behavior of this sequence?

### Related Questions:
1. What happens if \( U_0 \) is a negative number in this sequence?
2. How would the sequence behave if \( U_0 \) is large?
3. Can the sequence converge, or does it diverge in all cases?
4. What are the properties of quadratic recurrence relations in general?
5. How can we predict the behavior of the sequence without calculating every term?

### Tip:
Quadratic recurrences can grow very fast, so be cautious when dealing with large values early in the sequence.

Explore the recursive relation Un+1 = (Un + 10)(Un + 8) and learn how quadratic recurrence sequences evolve. This page provides detailed steps to analyze the recursive equation, suitable for students in Grades 10-12.

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