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Thread: Difference Equations

  1. #1
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    Difference Equations

    Here's my question:



    The answer to this probelm is:

    $\displaystyle
    S_{n} = 0.076 S_{n-1}
    $

    I don't see how to get this answer, here's my attempt:

    Equation has to be of the form $\displaystyle S_{n} = k S_{n-1}$, where k is some constant. I don't understand how they've got k=0.076

    Any explanation would be helpful.
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  2. #2
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    Clearly this should be:

    $\displaystyle S_n = 0.976 \, S_{n-1}$

    so either you've mistyped it or the person who set the question mistyped it putting a zero in place of where the 9 should have been!
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  3. #3
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    Quote Originally Posted by the_doc View Post
    Clearly this should be:

    $\displaystyle S_n = 0.976 \, S_{n-1}$

    so either you've mistyped it or the person who set the question mistyped it putting a zero in place of where the 9 should have been!

    Oh, ok, but how did you get 0.976?
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  4. #4
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    Quote Originally Posted by Roam View Post
    Oh, ok, but how did you get 0.976?
    $\displaystyle S_{n} - S_{n-1} = -0.024S_{n-1} $
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  5. #5
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    I'm sorry but I can't follow what you've done, could you please explain a little more?
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  6. #6
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    Say after $\displaystyle (n-1)$ years the amount is $\displaystyle S_{n-1}$ so then after one more year you lose 2.4 % which is $\displaystyle 0.024$ of $\displaystyle S_{n-1}$ so

    the amount lost $\displaystyle = 0.024 S_{n-1}$ so then after $\displaystyle n$ years we have the amount $\displaystyle S_n$ given by the amount at the start of the year, $\displaystyle S_{n-1}$, minus the amount lost so

    $\displaystyle S_n = S_{n-1} - 0.024 S_{n-1}$,

    $\displaystyle = (1-0.024) S_{n-1}$,

    $\displaystyle = 0.976 S_{n-1}$.

    Any better?
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  7. #7
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    ... How did you get 0.976?
    100% - 2.4% = 97.6%
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