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Math Help - Geometric Series

  1. #1
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    Geometric Series

    The first term of a geometric series is 27, the last term is 8 and the sum of the series is 65. What is the common ratio and how many terms are there in the series?
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  2. #2
    MHF Contributor red_dog's Avatar
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    a_1=27

    a_n=8\Rightarrow a_1q^{n-1}=8\Rightarrow q^{n-1}=\frac{8}{27}

    S_n=65\Rightarrow a_1\frac{q^n-1}{q-1}=65\Rightarrow\frac{q^n-1}{q-1}=\frac{65}{27}

    q^n=q^{n-1}\cdot q=\frac{8}{27}q

    Then, \frac{\frac{8}{27}q-1}{q-1}=\frac{65}{27}\Rightarrow q=\frac{2}{3}

    \left(\frac{2}{3}\right)^{n-1}=\frac{8}{27}\Rightarrow n=4
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