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Math Help - Fibonacci series cancellation

  1. #1
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    Question Fibonacci series cancellation

    I am currently re-viewing some text books of a course I took a few years ago and have come across this Fibonacci relation in the form Fn = Fn+2 – Fn+1

    F0 = F2 - F1
    F1 = F3 - F2
    F2 = F4 - F3
    …
    Fn = Fn+2 – Fn+1

    Adding LHS together we get LHS = F0 + F1 + F2 + … + Fn


    and RHS = F2 - F1 + F3 - F2 + F4 - F3 + … + Fn+2 – Fn+1

    And using telescoping cancellation the book states that the RHS = Fn+2 - 1

    I can only get the RHS to = Fn+2 – Fn+1 - 1 (Because F1 = 1). I am stuck with the – Fn+1 term of the RHS; nothing I can see cancels it out, yet clearly the book states that it has gone. Unfortunately the book doesn’t clearly show the cancellation in steps. Anyone here like to help, am I being very dumb???

    (sorry about the formatting; I subscripted all the numbers but it doesn't seem to cut and paste into this)
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  2. #2
    MHF Contributor
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    West Malaysia
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    Quote Originally Posted by muglyhacker View Post
    I am currently re-viewing some text books of a course I took a few years ago and have come across this Fibonacci relation in the form Fn = Fn+2 – Fn+1

    F0 = F2 - F1
    F1 = F3 - F2
    F2 = F4 - F3
    …
    Fn = Fn+2 – Fn+1

    Adding LHS together we get LHS = F0 + F1 + F2 + … + Fn


    and RHS = F2 - F1 + F3 - F2 + F4 - F3 + … + Fn+2 – Fn+1

    And using telescoping cancellation the book states that the RHS = Fn+2 - 1

    I can only get the RHS to = Fn+2 – Fn+1 - 1 (Because F1 = 1). I am stuck with the – Fn+1 term of the RHS; nothing I can see cancels it out, yet clearly the book states that it has gone. Unfortunately the book doesn’t clearly show the cancellation in steps. Anyone here like to help, am I being very dumb???

    (sorry about the formatting; I subscripted all the numbers but it doesn't seem to cut and paste into this)
    HI

    F_0=F_2-F_1

    F_1=F_3-F_2

    F_2=F_4-F_3

    .
    .
    .
    F_{n-1}=F_{n+1}-F_n

     F_{n}=F_{n+2}-F_{n+1}

    Just write one more before the last one , then you should be able to see clearly.
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  3. #3
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    Ah hah, thank you for that.
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