3Fn − Fn−2 = Fn+2, for n ≥ 3.

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The sequence of Fibonacci numbers, $F_0, F_1, F_2,F_3, . . .,$ are defined by the following equations:

$F_0=0,$ $F_1=1,$ $F_n=F_{n-1}+F_{n-2}, n\geq 2$

Then for $n\geq 3$

$F_{n+2}=F_{n+1}+F_n=F_n+F_{n-1}+F_n$

$=2F_n+F_n-F_{n-2}=3F_n-F_{n-2}$

Therefore

$3F_n-F_{n-2}=F_{n+2}, n\geq 3$

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