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Math Help - please help geometric sequence??????

  1. #1
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    please help geometric sequence??????

    consider the following geometric sequence 400, 320, 256, 204.8, ......

    i) what is the recurrance system that descibes this sequence?
    ( denote the sequence by xn, and its first term by x1.)

    ii) find the closed form for this sequence.

    iii) use the closed form from (ii) to find the tenth term of the sequence, giving your answer correct to 4 d.p.
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    Quote Originally Posted by toplad View Post
    consider the following geometric sequence 400, 320, 256, 204.8, ......

    i) what is the recurrance system that descibes this sequence?
    ( denote the sequence by xn, and its first term by x1.)

    ii) find the closed form for this sequence.

    iii) use the closed form from (ii) to find the tenth term of the sequence, giving your answer correct to 4 d.p.
    The common ratio is \frac{400}{320} = \frac{5}{4}.

    Each term therefore is x_{n+1} = x_nr = \frac{5}{4}x_n.

    So x_2 = \frac{5}{4}x_1

    x_3 = \frac{5}{4}x_2 = \frac{5}{4}\frac{5}{4}x_2 = \left(\frac{5}{4}\right)^2x_1

    x_4 = \frac{5}{4}x_3 = \frac{5}{4}\left(\frac{5}{4}\right)^2 x_2 = \left(\frac{5}{4}\right)^3x_1.


    So in terms of x_1,

    x_{n+1}=\left(\frac{5}{4}\right)^n x_n.
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    Quote Originally Posted by toplad View Post
    consider the following geometric sequence 400, 320, 256, 204.8, ......

    i) what is the recurrance system that descibes this sequence?
    ( denote the sequence by xn, and its first term by x1.)

    ii) find the closed form for this sequence.

    iii) use the closed form from (ii) to find the tenth term of the sequence, giving your answer correct to 4 d.p.

    Supposing that we had a series

    S_n = x_1 + x_1 r + x_1 r^2 + \dots x_1 r^n.

    Multiply this whole series by r, we get

    rS_n = x_1 r + x_1 r^2 + x_1 r^3 + \dots x_1 r^{n+1}.

    Subtract S_n from rS_n and we get

    rS_n - S_n = x_1 r^{n+1} - x_1

    S_n(r - 1) = x_1 (r^{n+1} - 1)

    S_n = \frac{x_1 (r^{n+1} - 1)}{r - 1}.


    So if x_1 = 400 and r = \frac{5}{4} the closed form for this series is

    S_n = \frac{400 \left(\frac{5}{4}^{n+1} - 1\right)}{\frac{5}{4} - 1}.

    S_n = \frac{400 \left(\frac{5}{4}^{n+1} - 1\right)}{\frac{1}{4}}

    S_n = 1600 \left(\frac{5}{4}^{n+1} - 1\right).
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    Tisk tisk Toplad, shouldn't be posting your Open University Mathematics coursework questions on forums
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    Quote Originally Posted by toplad View Post
    consider the following geometric sequence 400, 320, 256, 204.8, ......

    i) what is the recurrance system that descibes this sequence?
    ( denote the sequence by xn, and its first term by x1.)

    ii) find the closed form for this sequence.

    iii) use the closed form from (ii) to find the tenth term of the sequence, giving your answer correct to 4 d.p.
    Please pm me with clarification on whether these question are part of a graded assessment task.

    Thread closed.
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