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Math Help - GP inequality

  1. #1
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    GP inequality

    Dear Sir,
    I have some
    Difficulty in understanding the last part of the below problem and would be very grateful if you can show me how to do it.
    Thank you very much
    Best Regards
    Kingman


    A geometric progression has first term a (where a>0), and common ratio r.
    The sum of the first n terms is Sn and the sum to infinity is S. Given that S2 is twice the value of the fifth term, find the value of r (no problem up till here. and r is found to be - 0.648)

    Hence find the least value of n such that Sn is within 5% of S.

    Can you provide the geometrical significance and interpretation of the last question?
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  2. #2
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    As n increases, more and more terms are included, but the terms are smaller and smaller, so the partial sum Sn keeps getting larger, but slows down as it approaches S.

    What they're looking for is the first Sn that exceeds 0.95S.

    Since Sn=(1-r^n)/(1-r) and S=1/(1-r), you need to have Sn/S=1-r^n=0.95.

    - Hollywood
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