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Math Help - Algebraic deduction help

  1. #1
    Member
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    Algebraic deduction help

    Hi Guys,

    I need a little help with the algebra in this inequality. I have made a bit of mess of it.

    Given,

    <br />
\dfrac{100x^{n+1}}{1 - x^{n+1}} < \dfrac{k}{100}<br />

    Deduce, that n must exceed,

    <br />
\dfrac{log \frac{100+k}{k}}{log \frac{1}{x}} - 1<br />

    Thanks.

    EDIT: Not sure why the latex code is giving an error. I checked it in MikTex and it works fine?
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  2. #2
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    Quote Originally Posted by mathguy80 View Post
    EDIT: Not sure why the latex code is giving an error. I checked it in MikTex and it works fine?
    There seem to be problems with LaTeX in MHF after yesterday's shutdown.

    I got to the required conclusion under a couple of assumptions, but I have 100^2 instead of 100. First, if 1 - x^{n+1} > 0 and 100^2 + k > 0, then the initial inequality can be solved for x^{n+1} to get

    x^{n+1} < k / (100^2 + k).

    Then if x > 0, we have

    (1 / x)^{n+1} > (100^2 + k) / k.

    Taking log of both sides and dividing by log(1 / x) (which requires that log(1 / x) > 0, i.e., 1 / x > 1, i.e., x < 1, which accords with 1 - x^{n+1} > 0 above), you get the required inequality.
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  3. #3
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    Thanks @emakarov. I think the required answer may be wrong. Your solution looks good to me. I had not make the assumption to divide by log 1/x. Good to know, thanks.
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