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Math Help - If there exists a non zero term independent of x in the expansion of ..., then n...?

  1. #1
    Super Member fardeen_gen's Avatar
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    If there exists a non zero term independent of x in the expansion of ..., then n...?

    If there exists a non zero term independent of x in the expansion of [ x^2 - ((2)/(x)^3)]^n , then n can't be:

    A) 7
    B) 12
    C) 9
    D) 15

    More than one options may be correct.
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  2. #2
    Senior Member DeMath's Avatar
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    Sorry, I did not notice: "...then n can't be"

    {\left( {{x^2} - \frac{2}{{{x^3}}}} \right)^n} = \frac{1}{{{x^{3n}}}} {\left( {{x^5} - 2} \right)^n} = \frac{1}{{{x^{3n}}}}\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}}k  \\n  \\\end{array} } \right)} {x^{5\left( {n - k} \right)}}{\left( { - 2} \right)^k}.

    For your problem

    \frac{{{x^{5\left( {n - k} \right)}}}}{{{x^{3n}}}} = 1 \Leftrightarrow {x^{5\left( {n - k} \right)}} = {x^{3n}} \Leftrightarrow 5\left( {n - k} \right) = 3n \Leftrightarrow

    \Leftrightarrow 5k = 2n \Leftrightarrow k = \frac{2}{5}n \Rightarrow n = \left\{ {0;{\text{ }}5;{\text{ }}10;{\text{ }}15;{\text{ }} \ldots ;{\text{ }}5m} \right\}.

    So, right answers are:
    A) 7
    B) 12
    C) 9
    Last edited by DeMath; February 14th 2009 at 02:03 AM.
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  3. #3
    MHF Contributor red_dog's Avatar
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    The general term is (-1)^kC_n^k(x^2)^{n-k}\left(\frac{2}{x^3}\right)^k=(-1)^kC_n^k2^kx^{2n-5k}

    If the term is independent of x then 2n-5k=0\Rightarrow k=\frac{2n}{5}

    But k is integer, so the only option for n is 15.
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