Claim: $L_n=F_{n-1}+F_{n+1}$ for all $n >0$
Could someone please help me prove this? My professor mentioned it in class, but didn't show us how to prove it. I am just curious. The $L$ stands for the Lucas numbers and the $F$ stands for the Fibonacci numbers.
The exact statement should be:
$$L_n=F_{n-1}+F_{n+1}$$
Prove it by Induction.
$P(1)$ and $P(2)$ are easy to check.
Then $P(n-1), P(n) \Rightarrow P(n+1)$ is easy:
$$L_{n+1}=L_n+L_{n-1}=F_{n-1}+F_{n+1}+F_{n-2}+F_{n}=\\(F_{n+1}+F_{n})+(F_{n-1}+F_{n-2})=F_{n+2}+F_n$$