1. **Problem:** Consider the sequence defined by the recurrence relation $$a_{n+2} = 3a_{n+1} - 2a_n$$ with initial terms $$a_1 = 2$$ and $$a_2 = 5$$. Find an explicit formula for $$a_n$$. 2. **Formula and rules:** This is a linear homogeneous recurrence relation with constant coefficients. The characteristic equation is $$r^2 - 3r + 2 = 0$$. 3. **Solve the characteristic equation:** Factor the quadratic: $$r^2 - 3r + 2 = (r - 1)(r - 2) = 0$$ So the roots are $$r_1 = 1$$ and $$r_2 = 2$$. 4. **General solution:** Since roots are distinct, the general term is $$a_n = A \cdot 1^n + B \cdot 2^n = A + B \cdot 2^n$$ where $$A$$ and $$B$$ are constants determined by initial conditions. 5. **Apply initial conditions:** - For $$n=1$$: $$a_1 = A + 2B = 2$$ - For $$n=2$$: $$a_2 = A + 4B = 5$$ 6. **Solve the system:** Subtract first from second: $$(A + 4B) - (A + 2B) = 5 - 2 \Rightarrow 2B = 3 \Rightarrow B = \frac{3}{2}$$ Substitute back: $$A + 2 \cdot \frac{3}{2} = 2 \Rightarrow A + 3 = 2 \Rightarrow A = -1$$ 7. **Final explicit formula:** $$a_n = -1 + \frac{3}{2} \cdot 2^n$$ This formula allows you to compute any term of the sequence directly without recursion.