[Algebra]
Problem: Suppose that $r$ and $s$ are the two roots of the equation $F_{k} x^{2}+F_{k+1} x+F_{k+2}=0$, where $F_{n}$ denotes the $n^{\text {th }}$ Fibonacci number. Compute the value of $(r+1)(s+1)$.

Solution: $\quad$ Distributing, $(r+1)(s+1)=r s+(r+s)+1=\frac{F_{k+2}}{F_{k}}+\left(-\frac{F_{k+1}}{F_{k}}\right)+1=\frac{F_{k+2}-F_{k+1}}{F_{k}}+1=\frac{F_{k}}{F_{k}}+1=\mathbf{2}$.

Final answer: 2