Finding the Limit of a Fibonacci Series

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Discussion Overview

The discussion revolves around finding the limit of the average of the Fibonacci sequence's last digits as the number of terms approaches infinity. Participants explore the implications of the Fibonacci sequence's properties and the periodic nature of its last digits.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents the Fibonacci sequence and asks for the limit of the average of its terms.
  • Another participant questions the relevance of the last digits of Fibonacci numbers to the limit being discussed.
  • Some participants assert that the Fibonacci sequence itself does not have a limit due to its unbounded nature, while acknowledging that limits can be applied to sequences in calculus.
  • There is a mention of the periodicity of the last digits of Fibonacci numbers, suggesting that this periodicity plays a role in calculating the limit.
  • A later reply corrects the initial interpretation, clarifying that the limit should be considered in terms of the last digits rather than the Fibonacci numbers themselves.
  • One participant claims to have calculated the limit as 14/3, referencing the periodic nature of the last digits.
  • Another participant agrees with the calculation of 14/3 as correct.

Areas of Agreement / Disagreement

Participants express differing views on the nature of limits in relation to the Fibonacci sequence. While some agree on the periodicity of the last digits, the overall discussion reflects a lack of consensus on the interpretation of the limit itself.

Contextual Notes

The discussion includes assumptions about the periodicity of the last digits and the implications of averaging them, but these assumptions are not universally accepted or resolved.

Who May Find This Useful

Readers interested in mathematical sequences, limits in calculus, and properties of the Fibonacci sequence may find this discussion relevant.

tehno
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Let:
[tex]a_{1}=a_{2}=1;a_{n+2}=a_{n+1}+a_{n};n\geq 1[/tex]

Let [itex]f_{n}[/itex] be the last digit in decimal notation
of Fibonacci number [itex]a_{n}[/itex].
Find:

[tex]\lim_{n\to\infty}\frac{a_{1}+a_{2}+...+a_{n}}{n}[/tex]
 
Last edited:
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Can you explain?
 
What does fn have to do with anything?
 
By theory, there is no limit to Fibonacci, unless I'm mistaken.

The sequence wouldn't be a sequence if there was a limit.
 
chuckd1356 said:
By theory, there is no limit to Fibonacci, unless I'm mistaken.

The sequence wouldn't be a sequence if there was a limit.
That doesn't quite make sense. A fair part of Calculus courses is devoted to limits of sequences! Of course, the Fibonacci sequence is increasing without upperbound so it has no limit. But the question is about the nth partial sum divided by n. That's a whole different matter.
 
What the hell is anyone talking about here?
 
HallsofIvy said:
That doesn't quite make sense. A fair part of Calculus courses is devoted to limits of sequences! Of course, the Fibonacci sequence is increasing without upperbound so it has no limit. But the question is about the nth partial sum divided by n. That's a whole different matter.

That's what I was getting at, thanks for clarifying!
 
The question is about the final digits, which are periodic with period 60. The sum of the 60 values is ***, so the average value at the limit is ***/60.

(It's not hard to calculate this, so I left it as an exercise. I can check it if you think you have an answer.)
 
correction (+ solution)

Let:
[tex]a_{1}=a_{2}=1;a_{n+2}=a_{n+1}+a_{n};n\geq 1[/tex]

Let [itex]f_{n}[/itex] be the last digit in decimal notation
of Fibonacci number [itex]a_{n}[/itex].

Find:

[tex]\lim_{n\to\infty}\frac{f_{1}+f_{2}+...+f_{n}}{n}[/tex]

My apology for the confusion I made.


EDIT:
Yes the key for the solution is "***/60".
IOW ,[itex]f_{1}=f_{61},f_{2}=..etc.[/itex]
I get:
[tex]\lim_{n\to\infty}\frac{f_{1}+f_{2}+...+f_{n}}{n}=\frac{14}{3}[/tex]
 
Last edited:
  • #10
Yes, 14/3 is right.
 

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