1. We are given a recurrence relation $u_{n} = 3u_{n-1} - 1$ with initial term $u_1 = -1$. We want to find the general term $u_n$. 2. The recurrence is linear and non-homogeneous. First, solve the associated homogeneous recurrence $u_n^h = 3u_{n-1}^h$. 3. The characteristic equation is $r = 3$, so the homogeneous solution is $u_n^h = A \cdot 3^{n-1}$ where $A$ is a constant. 4. Next, find a particular solution $u_n^p$ to the non-homogeneous recurrence. Since the non-homogeneous term is constant $-1$, try a constant solution $u_n^p = C$. 5. Substitute $u_n^p = C$ into the recurrence: $$C = 3C - 1$$ 6. Solve for $C$: $$C - 3C = -1$$ $$\cancel{C} - 3\cancel{C} = -1$$ $$-2C = -1$$ $$C = \frac{1}{2}$$ 7. The general solution is: $$u_n = u_n^h + u_n^p = A \cdot 3^{n-1} + \frac{1}{2}$$ 8. Use the initial condition $u_1 = -1$ to find $A$: $$u_1 = A \cdot 3^{0} + \frac{1}{2} = A + \frac{1}{2} = -1$$ 9. Solve for $A$: $$A = -1 - \frac{1}{2} = -\frac{3}{2}$$ 10. Therefore, the explicit formula for $u_n$ is: $$u_n = -\frac{3}{2} \cdot 3^{n-1} + \frac{1}{2}$$ This formula allows you to compute any term $u_n$ of the sequence.