1. **Stating the problem:** We are given the recurrence relation $$U_{n+1} = 8U_n - \frac{8}{U_n} + 2$$ and need to analyze or solve it. 2. **Understanding the recurrence:** This is a nonlinear recurrence relation because of the term $\frac{8}{U_n}$. Such recurrences often require substitution or special techniques. 3. **Rewrite the recurrence:** Write it clearly as $$U_{n+1} = 8U_n - \frac{8}{U_n} + 2$$ 4. **Attempt substitution:** Let us try to simplify by multiplying both sides by $U_n$ to remove the fraction: $$U_n U_{n+1} = 8U_n^2 - 8 + 2U_n$$ 5. **No immediate simplification:** The relation is nonlinear and does not simplify easily. Without initial conditions or further instructions, we can only express the recurrence as given. 6. **Summary:** The recurrence relation is nonlinear and involves both $U_n$ and $\frac{1}{U_n}$. Solving it explicitly requires additional information or methods beyond the scope here. **Final answer:** The recurrence relation is $$U_{n+1} = 8U_n - \frac{8}{U_n} + 2$$ as given.