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Math Help - Fractions and indices

  1. #1
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    Fractions and indices

    this is a problem from a previous years MAOL math competition Matemaattisten Aineiden Opettajien Liitto MAOL ry : Etusivu_ala


    \frac{1}{2002^{-10}}+\frac{1}{2002^{-9}}\mbox{...}+\frac{1}{2002^0}\mbox{...}+\frac{1}{  2002^{9}}+\frac{1}{2002^{10}}


    Could someone please help me solve this? I tried various rules for indices, but noticed that the ones we have been taught work only for multiplication and division.





    Thank you in advance!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Coach View Post
    this is a problem from a previous years MAOL math competition Matemaattisten Aineiden Opettajien Liitto MAOL ry : Etusivu_ala


    \frac{1}{2002^{-10}}+\frac{1}{2002^{-9}}\mbox{...}+\frac{1}{2002^0}\mbox{...}+\frac{1}{  2002^{9}}+\frac{1}{2002^{10}}


    Could someone please help me solve this? I tried various rules for indices, but noticed that the ones we have been taught work only for multiplication and division.

    \frac{1}{2002^{-10}}+\frac{1}{2002^{-9}}\mbox{...}+\frac{1}{2002^0}\mbox{...}+\frac{1}{  2002^{9}}+\frac{1}{2002^{10}}

    ....... <br />
\ \ \ \ =\frac{1}{2002^{-10}}\left[ 1+\frac{1}{2002} + ... + \frac{1}{2002^{20}} \right]<br />

    The expression inside the square brackets is a finite geometric series whose sum is:

    <br />
\left[ 1+\frac{1}{2002} + ... + \frac{1}{2002^{20}} \right]= \frac{1-2002^{-21}}{1-2002^{-1}}<br />

    So:

    \frac{1}{2002^{-10}}+\frac{1}{2002^{-9}}\mbox{...}+\frac{1}{2002^0}\mbox{...}+\frac{1}{  2002^{9}}+\frac{1}{2002^{10}}

    ....... <br />
\ \ \ \ =\frac{1}{2002^{-10}}\left[ 1+\frac{1}{2002} + ... + \frac{1}{2002^{20}} \right]<br />

    ....... <br />
\ \ \ \ =2002^{10} \frac{1-2002^{-21}}{1-2002^{-1}}<br />

    Which you can simplify further if required

    RonL
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  3. #3
    MHF Contributor red_dog's Avatar
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    The terms form a geometric progression with ratio \displaystyle\frac{1}{2002} and the sum contain 21 terms.
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