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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.

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  • $\begingroup$ Have you tried anything? $\endgroup$ – Ahaan S. Rungta Nov 23 '13 at 3:14
  • $\begingroup$ I don't even know where to start. $\endgroup$ – A Glenn Nov 23 '13 at 3:17
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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:

$$L_{n+1}=L_n+L_{n-1}=F_{n-1}+F_{n+1}+F_{n-2}+F_{n}=\\(F_{n+1}+F_{n})+(F_{n-1}+F_{n-2})=F_{n+2}+F_n$$

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