TITLE: Fibonacci identity: $f_{n+1}f_{n-1} = f_n^2 + (-1)^n$
QUESTION [15 upvotes]: How do I see that
$$f_{n+1}f_{n-1} = f_n^2 + (-1)^n$$
for $n \ge 2$, where $f_1 = 1$, $f_2 = 1$, and $f_{n+2} = f_{n+1} + f_n$ for $n \in \mathbb{N}$?

REPLY [2 votes]: This is not something that you "see", but rather something that you discover while fiddling with Fibonacci numbers and asking yourself "I wonder if there is a relationship between ... and ...?".
The way that you see it is just by creating a table of some quantities that you have been "fiddling" with
$$\begin{array}{cccc}
k&f_k&f_{k-1}f_{k+1}&f_{k}^2\\
0&0&&\\
1&1&0&1\\
2&1&2&1\\
3&2&3&4\\
4&3&10&9\\
5&5&24&25\\
6&8&65&64\\
7&13&168&169\\
8&21&442&441\\
9&34&1155&1156\\
10&55&&\\
\end{array}
$$
It's not difficult to see the pattern.