Find (with proof) the length of the longest arithmetic subsequence of the Fibonacci sequence.

$\small\textbf{Source:}$ Self-made, but probably well-known by now.

## Arithmetic series in Fibonacci

- Phlembac Adib Hasan
**Posts:**1016**Joined:**Tue Nov 22, 2011 7:49 pm**Location:**127.0.0.1-
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### Arithmetic series in Fibonacci

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### Re: Arithmetic series in Fibonacci

I'm assuming that by 'subsequence' you mean the terms have to be consecutive terms of Fibo-Seq.

If there are at least $3$ terms then $F_{n+2}-F_{n+1}=F_{n+1}-F_{n}\Rightarrow 2F_{n+1}=F_n+F_{n+2}$ since common difference of consecutive terms is equal. Using $F_{n+2}=F_{n+1}+F_n$ this reduces to $F_{n+1}=2F_n$ which again reduces to $F_n=F_{n-1}$. So we must have $F_{n-1}=F_n=1$ and the longest subsequence is $1,2,3$.

If there are at least $3$ terms then $F_{n+2}-F_{n+1}=F_{n+1}-F_{n}\Rightarrow 2F_{n+1}=F_n+F_{n+2}$ since common difference of consecutive terms is equal. Using $F_{n+2}=F_{n+1}+F_n$ this reduces to $F_{n+1}=2F_n$ which again reduces to $F_n=F_{n-1}$. So we must have $F_{n-1}=F_n=1$ and the longest subsequence is $1,2,3$.

**- What is the value of the contour integral around Western Europe?**

- Zero.

- Why?

- Because all the poles are in Eastern Europe.

- Zero.

- Why?

- Because all the poles are in Eastern Europe.

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- Samiun Fateeha Ira
**Posts:**23**Joined:**Sat Aug 24, 2013 7:08 pm**Location:**Dhaka, Bangladesh

### Re: Arithmetic series in Fibonacci

But subsequence does not necessarily need to be of consecutive terms of the original sequence, does it?

### Re: Arithmetic series in Fibonacci

Clearly $F_{m}-F_{n}>F_{n+2}-F_{n}>F_{n}-1>F_{n}-F_{k}$ for all $m\geq (n+2), k\leq (n-1)$. This means $F_{n+1}$ is the only possible term which can be in an arithmetic sequence with $F_{n}$ and $F_{k}$ where $k<n$.Samiun Fateeha Ira wrote:But subsequence does not necessarily need to be of consecutive terms of the original sequence, does it?

Am I right?

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### Re: Arithmetic series in Fibonacci

A substring of a string is any portion cut off from the string, so... I'm not quite sure about what is meant here by subseq.Samiun Fateeha Ira wrote:But subsequence does not necessarily need to be of consecutive terms of the original sequence, does it?

**- What is the value of the contour integral around Western Europe?**

- Zero.

- Why?

- Because all the poles are in Eastern Europe.

- Zero.

- Why?

- Because all the poles are in Eastern Europe.

Revive the IMO marathon.

- Phlembac Adib Hasan
**Posts:**1016**Joined:**Tue Nov 22, 2011 7:49 pm**Location:**127.0.0.1-
**Contact:**

### Re: Arithmetic series in Fibonacci

In math (and programming), a sequence is called a subsequence of another sequence if all of its terms appear in the same order (but not necessarily consecutive) in the other sequence. Example: $F_1,F_3,F_4,F_{10}$ is a subsequence of the Fibonacci sequence but $F_2,F_1,F_4$ isn't.Nirjhor wrote:A substring of a string is any portion cut off from the string, so... I'm not quite sure about what is meant here by subseq.

@Sanzeed, yes you are. Your procedure proves there can be only one type of arithmetic subsequence, namely $F_n,F_{n+2},F_{n+3}$

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