Cassini's identity

Statement

Let $n$ be a positive integer, and let $F_m$ be the Fibonacci numbers. Then $$F_{n-1} F_{n+1} - F_n^2 = (-1)^n.$$

Remarks

Examples: for $n = 5$, $F_4 F_6 - F_5^2 = 3 \cdot 8 - 25 = -1$. For $n = 6$, $F_5 F_7 - F_6^2 = 5 \cdot 13 - 64 = 1$. A short proof takes determinants in the matrix identity $\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1} \end{pmatrix}$. Catalan's identity $F_n^2 - F_{n-r} F_{n+r} = (-1)^{n-r} F_r^2$ generalizes it. The case $r = 1$ is Cassini's identity.

Explanations

L1 human

The product of the two neighbours of a Fibonacci number differs from its square by exactly $1$. The sign alternates: $1 \cdot 3 - 2^2 = -1$, then $2 \cdot 5 - 3^2 = 1$, then $3 \cdot 8 - 5^2 = -1$.