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Math Help - find the period of the function

  1. #1
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    find the period of the function

    find the period of the function ....

    f(x)+f(x+1)=f(x+2)
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by inolas_cul View Post
    find the period of the function ....

    f(x)+f(x+1)=f(x+2)
    This relation does not define a function, you will have to provide more
    information before you will get a sensible answer without us guessing the
    real question.

    RonL
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  3. #3
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    For example, if f(x)=0 on R then their is not fundamental period
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    hey the ques is find the period of ...
    f(x)+f(x+1)=f(x=2)
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by inolas_cul View Post
    find the period of the function ....

    f(x)+f(x+1)=f(x+2)
    Quote Originally Posted by inolas_cul View Post
    hey the ques is find the period of ...
    F(x)+F(x+1)=f(x=2)
    Your second statement of the question is slightly different from the first. Does F(x) mean something in relation to f(x)?

    -Dan
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  6. #6
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    no it doesnt...its the same......it was a typin mistake i corrected it...as soon as i realized...
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  7. #7
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by inolas_cul View Post
    no it doesnt...its the same......it was a typin mistake i corrected it...as soon as i realized...
    Then I will echo CaptainBlack's reply. We need more information if we are to help you with this problem. I can't find a way to discover a form for the function, much less a period for it.

    -Dan
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  8. #8
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    hey tats d ques nothin beyond it...and it has a solution.....
    d ans is a multiple of 8....nothin else i kno abt the ans...
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by inolas_cul View Post
    hey the ques is find the period of ...
    f(x)+f(x+1)=f(x=2)
    And we are telling you that:

    f(x) + f(x+1) = f(x+2)

    does not determine a unique minimum period for f. In fact you have not even
    indicated the domain of f, for all we know it could be Z or R. I would usually
    assume that we were talking about functions on R for this type of question
    In fact the recurence does not even ensure that f is periodic.

    Now it may be the case that if f is a periodic solution (on either Z or R) then
    there is a minimum period that the solution can have, but that is not the question
    being asked.

    RonL
    Last edited by CaptainBlack; May 13th 2007 at 08:30 PM. Reason: remove erroneous solution
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  10. #10
    Forum Admin topsquark's Avatar
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    Perhaps it would help if you could tell us what class you are taking, what material (ie. chapter heading) this question originated from, what you were recently talking about in class, etc. Questions don't arise from a vacuum and evidently you are supposed to have a way to solve this thing. So if we know the topic it came from, maybe we can help you.

    -Dan
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  11. #11
    Grand Panjandrum
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    After wasting rather more time on this than I think its worth, I can tell
    you there are no periodic solutions to the recurrence:

    f(x+2)=f(x+1)+f(x)

    for f a function over Z, and the same proof shows that there aslo cannot
    be a periodic solution for f a function over R either.

    The solutions to this recurrence are generalised Fibinacci numbers and their
    absolute values grow without bound.

    RonL
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  12. #12
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    hey d topic is functions.....
    n d domain is R....nythin else u wanna kno abt d prob wid out which it cant b solved....
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  13. #13
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    wel can i kno moreabt d proof u r tokin abt...
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  14. #14
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    fibonacci is only fr int rt???
    2nd thin a fn need not hav sumthin lik sin cos root powers
    it can be defined for 1 n 2 n 3 n4such as it follows dat eqn
    and more dan dat .....if i say f(1)=-1 and f(2)=3 then f(3)=2 dat says it is not an ever increasin or decreasin fn
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  15. #15
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    A second order linear recurence like the one in question has a solution uniquely
    determined at all integer values of x by the values of f(1) and f(2).

    Now let

    f(n+2) = f(n+1) + f(n) ...(1)

    and use a trial solution f(n)=a^n, then:

    a^{n+2}=a^{n+1}+a^n,

    so:

    a^2 - a - 1=0.

    This has roots (1+sqrt(5))/2 and (1-sqrt(5))/2, so any solution of (1) can be
    written:

    f(n) = A[(1+sqrt(5))/2]^n + B[(1-sqrt(5))/2]^n

    and values of A and B can be found to satisfy any values we require for f(1)
    and f(2).

    Now (1+sqrt(5))/2 = g ~=1.62, and (1-sqrt(5))/2 ~= 0.62, so as n becomes
    large f(n) ~ A g^n. So the absolute value of f(n) goes to infinity as n goes
    to infinity. Thus there is no periodic function over the integers Z that satisfies
    the recurence (1).

    Now if f is a function over R which satisfies (1), at the very least f(x), x an
    integer must be bounded, but we have seen that it cannot be, so there are
    no functions over R which are periodic solutions of (1). (there is a get out clause
    here in that we could allow f to have singularities, which would then allow the
    function to increase without bound at integer values of its argument but with
    a forrest of infinities in each unit interval - as far as I can tell).

    RonL

    (Aside: there are always periodic solutions for functions over Z/nZ, in the
    case of Z/2Z, they are all of period 3)
    Last edited by CaptainBlack; May 14th 2007 at 03:33 AM.
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