Fibonacci numbers/Simpson formula/Exercise/Solution

The initial case for $n = 2$ is clear because of

f_{3} \cdot f_{1} - f_{2}^{2} = 2 \cdot 1 - 1^{2} = 1 = (- 1)^{2} .

We assume now that the statement is already known for some $n$ and we consider the statement for $n + 1$ . This follows from

\begin{matrix} f_{n + 2} f_{n} - f_{n + 1}^{2} & = (f_{n} + f_{n + 1}) f_{n} - f_{n + 1}^{2} \\ = f_{n}^{2} + f_{n + 1} f_{n} - f_{n + 1}^{2} \\ = f_{n}^{2} + f_{n + 1} (f_{n} - f_{n + 1}) \\ = f_{n}^{2} + f_{n + 1} (f_{n} - f_{n} - f_{n - 1}) \\ = f_{n}^{2} - f_{n + 1} f_{n - 1} \\ = (- 1) (f_{n + 1} f_{n - 1} - f_{n}^{2}) \\ = (- 1) (- 1)^{n} \\ = (- 1)^{n + 1} \end{matrix}