2fn-f(n-2) = fn+1 for n>3

The recursive definition for generating Fibonacci numbers and the Fibonacci sequence is:

$f_n = f_{n-1} + f_{n-2} \ where\ n\geq3$

Then

$f_{n+1} = f_{n} + f_{n-1}$

$f_{n-1}=f_{n+1}-f_{n}$

Substitute

$f_n = f_{n-1} + f_{n-2}=f_{n+1}-f_{n}+ f_{n-2}$

Therefore

$2f_n-f_{n-2} =f_{n+1}, n\geq3,\ True$