Catalan's Identity

Theorem
Let $F_k$ be the $k$th Fibonacci number.

Then:
 * $\displaystyle F_n^2 - F_{n-r} F_{n+r} = \left( -1 \right) ^ {n-r} F_r$.

Proof
From the initial definition of Fibonacci numbers, we have:
 * $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$

By Fibonacci Numbers in terms of Smaller Fibonacci Numbers:
 * $\displaystyle F_n = F_{n-r} F_{r-1} + F_{n-r+1} F_{r}$

Also:

Therefore: