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Math Help - Series and Sequences Last part

  1. #1
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    Series and Sequences Last part

    The sum of the first 4 terms of a geometric series is 30, and the sum of the infinite series is 32. Find the first 3 terms.

    For the geometric sequence a, ar, ar^2,…show that the sequence log a, log (ar), log (ar^2) is an arithmetic sequence

    Find a number which, when added to each of 2,6 13 gives three number in geometric sequence

    These are the last questions, I apologise for posting up so many, I just really need some help with these, thanks again.
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  2. #2
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    Quote Originally Posted by christina View Post
    The sum of the first 4 terms of a geometric series is 30, and the sum of the infinite series is 32. Find the first 3 terms.

    For the geometric sequence a, ar, ar^2,…show that the sequence log a, log (ar), log (ar^2) is an arithmetic sequence

    Find a number which, when added to each of 2,6 13 gives three number in geometric sequence

    These are the last questions, I apologise for posting up so many, I just really need some help with these, thanks again.
    For the second part

    \log{(ar)} = \log{a} + \log{r}

    \log{(ar^2)} = \log{a} + \log{r} + \log{r}

    \log{(ar^3)} = \log{a} + \log{r} + \log{r} + \log{r}.


    I think it should be obvious that you have an arithmetic sequence with t_1 = \log{a} and d = \log{r}.
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    Quote Originally Posted by christina View Post
    The sum of the first 4 terms of a geometric series is 30, and the sum of the infinite series is 32. Find the first 3 terms.

    For the geometric sequence a, ar, ar^2,…show that the sequence log a, log (ar), log (ar^2) is an arithmetic sequence

    Find a number which, when added to each of 2,6 13 gives three number in geometric sequence

    These are the last questions, I apologise for posting up so many, I just really need some help with these, thanks again.
    For the first part:

    S_n = \frac{a(r^n - 1)}{r - 1}

    so S_4 = 30 = \frac{a(r^4 - 1)}{r - 1}.

    Call this equation (1).


    S_{\infty} = 32 = \frac{a}{1 - r}

    Call this equation (2).


    \frac{(1)}{(2)} gives

    1 - r^4 = \frac{30}{32}

    r^4 = \frac{1}{16}

    r = \frac{1}{2}.


    Since you know 32 = \frac{a}{1 - r}

    32 = \frac{a}{1 - \frac{1}{2}}

    32 = \frac{a}{\frac{1}{2}}

    32 = 2a

    a = 16.


    Since the first term is 16 and the ratio is \frac{1}{2}, the first three terms are

    16, 8, 4.
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