1. **Problem:** Solve the recurrence relation $$S(n) - 2S(n - 1) + S(n - 2) = 0$$ with initial conditions $$S(0) = 9$$ and $$S(1) = 10$$. 2. **Formula and approach:** This is a linear homogeneous recurrence relation with constant coefficients. The characteristic equation is: $$r^2 - 2r + 1 = 0$$ 3. **Solve characteristic equation:** $$r^2 - 2r + 1 = (r - 1)^2 = 0$$ This has a repeated root $$r = 1$$. 4. **General solution for repeated root:** For repeated root $$r$$, the solution is: $$S(n) = (A + Bn)r^n = A + Bn$$ 5. **Use initial conditions:** - For $$n=0$$: $$S(0) = A = 9$$ - For $$n=1$$: $$S(1) = A + B = 10$$ 6. **Find constants:** $$A = 9$$ $$9 + B = 10 \implies B = 1$$ 7. **Final solution:** $$\boxed{S(n) = 9 + n}$$ This means the sequence increases by 1 each step starting from 9. \n\n**Note:** The user message contains multiple problems, but per instructions, only the first problem is solved here.