- #1

evinda

Gold Member

MHB

- 3,836

- 0

I want to show that $L_{n+1}+L_{n-1}=5 F_n$ for $n \geq 2$ and conclude that $5 \nmid L_n$ for $n \geq 1$.

I have tried the following:

$L_{n+1}+L_{n-1}=F_n+F_{n+2}+F_{n-2}+F_n=F_n+F_{n+1}+F_n+F_n-F_{n-1}+F_n=4F_n+F_{n+1}-F_{n-1}=4F_n+F_n+F_{n-1}-F_{n-1}=5F_n$.

But how do we deduce that $5 \nmid L_n$ for $n \geq 1$ ? (Thinking)