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Math Help - Finding the recursive definition for an integer sequence

  1. #1
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    Finding the recursive definition for an integer sequence

    I'm not sure I fully understand how to do a recursive definition for an integer sequence containing an exponent.

    a_n = 4^n + 12; \forall n \geq 0

    I can figure out that:

    4^0 + 12 = 13
    4^1 + 12 = 16
    4^2 + 12 = 28
    4^3 + 12 = 76
    4^4 + 12 = 268


    which equates to (mutiples of 4) x 12...but I'm not sure how to finish this to make it a recursive definition. The question I'm trying to answer has different values, my hope is to learn how to do this in general and not the specific question I'm required to answer.

    Thank you.
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Finding the recursive definition for an integer sequence

    You could write:

    a_{n}=4^{n}+12

    a_{n+1}=4^{n+1}+12

    Now, subtract the former from the latter to get a recursion (difference equation).
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  3. #3
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    Re: Finding the recursive definition for an integer sequence

    Quote Originally Posted by Kevmck View Post
    I'm not sure I fully understand how to do a recursive definition for an integer sequence containing an exponent.
    a_n = 4^n + 12; \forall n \geq 0
    I can figure out that:
    4^0 + 12 = 13
    4^1 + 12 = 16
    4^2 + 12 = 28
    4^3 + 12 = 76
    4^4 + 12 = 268
    which equates to (mutiples of 4) x 12...but I'm not sure how to finish this to make it a recursive definition. The question I'm trying to answer has different values, my hope is to learn how to do this in general and not the specific question I'm required to answer.
    I do not understand what you are asking.
    What do you want done here?
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    Re: Finding the recursive definition for an integer sequence

    Quote Originally Posted by MarkFL2 View Post
    You could write:

    a_{n}=4^{n}+12

    a_{n+1}=4^{n+1}+12

    Now, subtract the former from the latter to get a recursion (difference equation).
    Thank you for the quick response.

    I'm guessing here, but that would look something like
      4^n^+^1 + 12
    - 4^n + 12

    And the only answer I seem to see right now is wrong:
    4^n^+^1 - 4^n = 1
    but that doesn't seem like it makes any sense...
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    Re: Finding the recursive definition for an integer sequence

    Quote Originally Posted by Plato View Post
    I do not understand what you are asking.
    What do you want done here?
    The question asks me to give the recursive definition for the integer sequence. The only example I have to work off of is:
    a_n = 4n - 1, n \geq 1
    a_n = 4n - 1
    a_1_0_0 = 4(100) - 1 = 399
    a_n_-_1 = 4(n-1) - 1 = 4n - 5
    a_n_-_1 = a_n - 4
    a_n = a_n_-_1 + 4; a_1 = 3; n \geq 2
    Last edited by Kevmck; November 19th 2012 at 01:01 PM.
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  6. #6
    MHF Contributor MarkFL's Avatar
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    Re: Finding the recursive definition for an integer sequence

    What I am suggesting would give you:

    a_{n+1}=a_n+3\cdot4^n where a_0=13
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    Re: Finding the recursive definition for an integer sequence

    I thank you for the explanation, but I must be doing something wrong. When I use that formula I seem to be getting different answers.
    a_1_+_1 = 13 + 3 \cdot 4^1 where a_1 = 25
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  8. #8
    MHF Contributor MarkFL's Avatar
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    Re: Finding the recursive definition for an integer sequence

    a_{n+1}=4^{n+1}+12

    a_{n}=4^{n}+12

    Subtracting the bottom from the top, we find:

    a_{n+1}-a_{n}=4^{n+1}-4^{n}+12-12

    a_{n+1}-a_{n}=4^{n}(4-1)

    a_{n+1}=a_{n}+3\cdot4^{n}

    Now, from the definition you provided, we find:

    a_{0}=4^0+12=13

    Hence, using the recursion we have found we get:

    a_{1}=16
    a_{2}=28
    a_{3}=76
    a_{4}=268
    Thanks from Kevmck
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    Re: Finding the recursive definition for an integer sequence

    Thank you MarkFL2. I'm starting to understand this now. I appreciate your help.
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    Re: Finding the recursive definition for an integer sequence

    Finding an explicit formula for a_n when a_n is defined by recursion is a meaningful problem. It allows calculating a_n directly, without going through a_1,\dots,a_{n-1}. The converse problem given an explicit formula, find the recursive definition does not make much sense, or, rather, it is trivial. If a_n=f(n), then a_n=g(n,a_{n-1}) where g(x,y)=f(x), i.e., g does not use the second argument. In this particular case, the recursive definition a_{n}=a_{n-1}+3\cdot4^{n-1} is not even simpler than the explicit formula a_{n}=4^{n}+12.

    Edit 1: The converse problem would make sense if in a_n=g(n,a_{n-1}) the function g can only depend on a_{n-1} but not n. This restricted form of recursion is sometimes called iteration. But in this case a_{n} is not a function of a_{n-1}, i.e., the sequence a_n cannot be defined using iteration.

    Edit 2: To be more precise, every sequence a_n defined by (primitive) recursion can also be defined by iteration, but the definition requires an intermediate sequence: first one defined b_n by iteration and then a_n = h(b_n) for some function h.
    Last edited by emakarov; November 20th 2012 at 12:14 AM.
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    Re: Finding the recursive definition for an integer sequence

    Quote Originally Posted by MarkFL2 View Post
    a_{n+1}=4^{n+1}+12

    a_{n+1}-a_{n}=4^{n+1}-4^{n}+12-12

    a_{n+1}-a_{n}=4^{n}(4-1)

    a_{n+1}=a_{n}+3\cdot4^{n}
    Really confused how you made these transitions. Can you help me understand?
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  12. #12
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    Re: Finding the recursive definition for an integer sequence

    Nevermind, I just realized that if you subtract 4n from 4*4n you get 3*4n

    Sorry its been a while
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