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Thread: 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.

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

    I can figure out that:

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


    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:

    $\displaystyle a_{n}=4^{n}+12$

    $\displaystyle a_{n+1}=4^{n+1}+12$

    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.
    $\displaystyle a_n = 4^n + 12; \forall n \geq 0$
    I can figure out that:
    $\displaystyle 4^0 + 12 = 13$
    $\displaystyle 4^1 + 12 = 16$
    $\displaystyle 4^2 + 12 = 28$
    $\displaystyle 4^3 + 12 = 76$
    $\displaystyle 4^4 + 12 = 268$
    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:

    $\displaystyle a_{n}=4^{n}+12$

    $\displaystyle a_{n+1}=4^{n+1}+12$

    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
    $\displaystyle 4^n^+^1 + 12$
    $\displaystyle - 4^n + 12$

    And the only answer I seem to see right now is wrong:
    $\displaystyle 4^n^+^1 - 4^n = 1$
    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:
    $\displaystyle a_n = 4n - 1, n \geq 1$
    $\displaystyle a_n = 4n - 1$
    $\displaystyle a_1_0_0 = 4(100) - 1 = 399$
    $\displaystyle a_n_-_1 = 4(n-1) - 1 = 4n - 5$
    $\displaystyle a_n_-_1 = a_n - 4$
    $\displaystyle a_n = a_n_-_1 + 4; a_1 = 3; n \geq 2$
    Last edited by Kevmck; Nov 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:

    $\displaystyle a_{n+1}=a_n+3\cdot4^n$ where $\displaystyle a_0=13$
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  7. #7
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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.
    $\displaystyle a_1_+_1 = 13 + 3 \cdot 4^1$ where $\displaystyle a_1 = 25$
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  8. #8
    MHF Contributor MarkFL's Avatar
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    Re: Finding the recursive definition for an integer sequence

    $\displaystyle a_{n+1}=4^{n+1}+12$

    $\displaystyle a_{n}=4^{n}+12$

    Subtracting the bottom from the top, we find:

    $\displaystyle a_{n+1}-a_{n}=4^{n+1}-4^{n}+12-12$

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

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

    Now, from the definition you provided, we find:

    $\displaystyle a_{0}=4^0+12=13$

    Hence, using the recursion we have found we get:

    $\displaystyle a_{1}=16$
    $\displaystyle a_{2}=28$
    $\displaystyle a_{3}=76$
    $\displaystyle a_{4}=268$
    Thanks from Kevmck
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  9. #9
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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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  10. #10
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    Re: Finding the recursive definition for an integer sequence

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

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

    Edit 2: To be more precise, every sequence $\displaystyle a_n$ defined by (primitive) recursion can also be defined by iteration, but the definition requires an intermediate sequence: first one defined $\displaystyle b_n$ by iteration and then $\displaystyle a_n = h(b_n)$ for some function h.
    Last edited by emakarov; Nov 20th 2012 at 12:14 AM.
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  11. #11
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    Re: Finding the recursive definition for an integer sequence

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

    $\displaystyle a_{n+1}-a_{n}=4^{n+1}-4^{n}+12-12$

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

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