If $F_n<a<F_{n+1}<b<F_{n+2}$ for $n\ge4$,establish that the sum $a+b$ cannot be a Fibonacci number

Question

If $F_n<a<F_{n+1}<b<F_{n+2}$ for $n\ge4$,establish that the sum $a+b$ cannot be a Fibonacci number.

I'm thinking of showing that $a+b$ cannot be written as the sum of two preceeding terms.But don't know HOW?

Please tell me how can i attain the desired result...

Answer 1

Yes, you are on the right track. We show that $a+b$ is strictly in between two consecutive Fibonacci numbers: $$F_{n+2}=F_{n}+F_{n+1}<a+b<F_{n+1}+F_{n+2}=F_{n+3}.$$