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Math Help - Fibonacci Sequence question

  1. #1
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    Fibonacci Sequence question

    Hey all,

    Im hoping you guys could point me in the right direction after getting myself rather lost with a question on my assignment. I am not putting the actual question as i would like to work it out myself. here is a question i have made up that is similar with what i think to be correct. Could anyone verify this or point me in the right direction?

    Thanks!

    Let Fn be the Fibonacci sequence.

    Use the Fibonacci recurrence relation to express Fn+3 in terms of Fn+1 and Fn. Hence show that Fn+1 = 1/2(Fn+5 - Fn) for n=0,1,2,...

    the next question i have is the same but with Fn+1 = 1/2(Fn+5 + Fn)

    then i have to decide wether the formulas would remain true if the sequence Fn were replaced by a sequence with the same recurrence relation as the Fibonacci sequence but with different initial terms, then justify.

    So if i look just at the 1st one at the moment, i think that i would get something like this;

    Fn+1 = 1/2(Fn+5 - Fn) for n=0,1,2,...

    Fn = 1/2(Fn+1 - Fn+1)
    Fn = 1/2(Fn+2 + Fn+1)

    if that is not correct, how do i work this out properly as im rather stumped :-S
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  2. #2
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    Re: Fibonacci Sequence question

    Quote Originally Posted by Arkious View Post
    Hence show that Fn+1 = 1/2(Fn+5 - Fn) for n=0,1,2,...
    but this is not true.

    Presumably you meant F_{n+1}=\frac{1}{2}(F_{n+3}-F_n).
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  3. #3
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    Re: Fibonacci Sequence question

    Yeah that's the question I actually have, I just didn't want to ask the question I have been given. I'm not 100% with Fibonacci yet. But if you could explain how I would complete the question you have mentioned, I would much much appreciate it
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    Re: Fibonacci Sequence question

    Dont know what you have asked above, the notation is confusing, It gives 2 different expressions for Fn+1 alone. I will try to give you the proof for Fn+1 = 1/2(Fn+3 - Fn).

    Fn+3 = Fn+2 + Fn+1
    = (Fn+1 + Fn) + Fn+1
    = 2Fn+1 + Fn.

    Just rearrange terms here.

    Salahuddin
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  5. #5
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    Re: Fibonacci Sequence question

    Yeah that's the problem, it's very confusing!

    If i didn't have to show that Fn+1 = 1/2(Fn+3 - Fn) for n=0,1,2,... I would have been able to do it.

    Thanks for your input :-) anyone got any ideas though? I'm no further forward.
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  6. #6
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    Re: Fibonacci Sequence question

    Did you try to express F_{n+3} in terms of F_{n+1} and F_n ?

    You could start by writing F_{n+3} in terms of F_{n+2} and F_{n+1}.
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  7. #7
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    Re: Fibonacci Sequence question

    firstly thanks for your replies,

    right, this is what i have done so far...

    in Fibonacci terms;
    Fn+3 = Fn+2 + Fn+1
    Fn+2 = (Fn+2 + Fn)
    Fn+1 = (Fn+2 - Fn)
    Fn = (Fn+2 -Fn+1)

    (if thats correct)

    But then i need to express Fn+1 = 1/2(Fn+3 - Fn)
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  8. #8
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    Re: Fibonacci Sequence question

    in Fibonacci terms;
    Fn+3 = Fn+2 + Fn+1
    Fn+2 = (Fn+1 + Fn)

    Correct for these two, but now, get rid of the Fn+2 in the first equation, since we do not have any Fn+2 in our formula. Substitute your Fn+2 = Fn + Fn+1 there, and rearrange terms.

    Salahuddin
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  9. #9
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    Re: Fibonacci Sequence question

    but this still doesnt make sense... how do i then show that Fn+1 = 1/2(Fn+3 - Fn).

    surely above is all wrong?
    Last edited by Arkious; November 1st 2012 at 04:23 AM.
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  10. #10
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    Re: Fibonacci Sequence question

    Fn+3 = Fn+2 + Fn+1
    (2) = (1) + (1)
    Fn+2 = Fn+1 + Fn
    (1) = (1) + (0)
    Fn+1 = Fn - Fn-1
    (1) = (0) - (-1)
    Fn = Fn-1 - Fn-2
    (0) = (-1) - (-1)

    Thats proved that those fibonacci terms are correct, but then how do i then show Fn+1 = 1/2(Fn+3 - Fn)?
    Last edited by Arkious; November 1st 2012 at 04:26 AM.
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  11. #11
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    Re: Fibonacci Sequence question

    Guys, here is the steps.

    Fn+3 = Fn+2 + Fn+1
    But since Fn+2 = Fn+1 + Fn

    Fn+3 = Fn+2 + Fn+1
    = (Fn+1 + Fn) + Fn+1
    = 2Fn+1 + Fn

    Remove Fn, but subtracting it both sides,

    Fn+3 - Fn = 2Fn+1.

    In other words,

    Fn+1 = 1/2(Fn+3 - Fn). This is the same style for many other problems, get it?

    Salahuddin
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  12. #12
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    Re: Fibonacci Sequence question

    Fn+1 = 1/2(Fn+3 - Fn)
    (5) = 1/2 ((13) - (3))
    (5) = 1/2 (10)

    am i supposed to find the solution for Fn then use that to find Fn+3?

    I have tried multiple ways and cant find a solution that works for more than one number :-(
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  13. #13
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    Re: Fibonacci Sequence question

    Hi Arkious, could you be more specific, I am not able to understand what you are saying. (Basically if you have Fn and Fn+1, you can find Fn+3. But why are you finding the solutions, what is the actual problem?).

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  14. #14
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    Re: Fibonacci Sequence question

    Let Fn be the Fibonacci sequence.

    Use the Fibonacci recurrence relation to express Fn+3 in terms of Fn+1 and Fn. Hence show that Fn+1 = 1/2(Fn+3 - Fn) for n=0,1,2,...


    That is the question i have to answer. My issue is how to show that Fn+1 = 1/2(Fn+3 - Fn).

    I dont think that these are of any use for the question i have to answer.

    Fn+3 = Fn+2 + Fn+1
    Fn+2 = Fn+1 + Fn
    Fn+1 = Fn - Fn-1
    Fn = Fn-1 - Fn-2

    All i need to know is how to answer that question and i think that what i have done so far has confused me more. I just don't understand how im supposed to show that Fn+1 = 1/2(Fn+3 - Fn) for Fn+3 & Fn+1
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  15. #15
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    Re: Fibonacci Sequence question

    Quote Originally Posted by Arkious View Post

    Fn+3 = Fn+2 + Fn+1
    Fn+2 = Fn+1 + Fn
    These are what you need.

    Use the second one to substitute for Fn+2 in the first.

    Fn+3 = Fn+2 + Fn+1

    becomes

    Fn+3 = _?_+_?_ + Fn+1.
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