Answer to Question #277054 in Discrete Mathematics for Larvyn Mata

Question

3Fn − Fn−2 = Fn+2, for n ≥ 3.

Answer 1

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

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