Doubt in a property of the Fibonacci Series.

Question

I came across this question in a book where they asked me to prove that there are exactly four terms such that $F_{F_n}= F_n$. Well, I think that this is false and that there are exactly three. I have proof for my statement. Please tell me whether my proof is wrong, or the question is.

It can be clearly seen that $F_0=0$, $F_1=1$ and $F_5=5$.For all $n>5$, we know can prove that $F_n> n$ as $F_n= F_{n-1} + F_{n-2}$ where $F_{n-1}\geq n-1$ and $F_{n-2}>1$. Therefore their sum, $F_n>n$. Is this proof correct, or is there a fourth term.

score 2 · Accepted Answer · answered May 02 '16 at 08:05

2

is there a fourth term

Yes : $$F_{F_2}=F_2$$

answered May 02 '16 at 08:05

mathlove

139,939

1

It's rather easy to miss that it's not true that $$F_{F_n}=F_n \Leftrightarrow F_n=n$$ – S.C.B. May 02 '16 at 08:07
oh yes, i din't think of it like that....thank you.... – Coherent Sheaf May 02 '16 at 08:07
@MXYMXY....well $F_{F_n}=F_n⇔F_n=n$ is true...the thing it that if i take it as, $F_1=1$, there are two cases for which I can get that 1. – Coherent Sheaf May 02 '16 at 08:12

Doubt in a property of the Fibonacci Series.

1 Answers1