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Math Help - Sequences and series

  1. #1
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    Sequences and series

    The second term of a geometric series is 24 and the fifth term is 3.
    a. Show that the common ratio of the series is 1/2.
    b. Find the first term of the series.
    c. Find the sum to infinity of the series.
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  2. #2
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    Quote Originally Posted by ansonbound View Post
    The second term of a geometric series is 24 and the fifth term is 3.
    a. Show that the common ratio of the series is 1/2.
    b. Find the first term of the series.
    c. Find the sum to infinity of the series.
    That's pretty straightforward- if you know the definitions. The "nth" term of a geometric series with first term a and common ratio r is ar^{n-1}. Knowing that the second term is 24 tells you that ar^{2-1}= ar= 24 and knowing that the 5th term is 3 tells you that ar^{5-1}= ar^4= 3. That gives you two equations to solve for a and r. In particular, if you divide the second equation by the first,
    \frac{ar^4}{ar}= r^3= \frac{3}{24}= \frac{1}{8}

    Once you know r, it is easy to use ar= 24 to find a.

    And, once you know both a and r, the "sum to infinity" is just
    \frac{a}{1- r}
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  3. #3
    A riddle wrapped in an enigma
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    Quote Originally Posted by ansonbound View Post
    The second term of a geometric series is 24 and the fifth term is 3.
    a. Show that the common ratio of the series is 1/2.
    b. Find the first term of the series.
    c. Find the sum to infinity of the series.
    Hi ansonbound,

    ____, 24, ____, ____, 3

    Using a_n=a_1 \cdot r^{n-1}, we can verify that the common ratio is 1/2

    a_n=3
    a_1=24
    n=4

    3=24 \cdot r^{4-1}

    \frac{1}{8}=r^3

    r=\frac{1}{2}

    Working to the left of 24, the first term if found by multiplying 24 by 2. Thus, the first term is 48.

    The sum S of an infinite geometric series with -1 < r < 1 is given by:

    S=\frac{a_1}{1-r}

    Since r=\frac{1}{2}, and since \left|\frac{1}{2}\right| <1, the sum exists.

    S=\frac{48}{1-\frac{1}{2}}

    S=\frac{48}{\frac{1}{2}}

    S=96
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  4. #4
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    THANKS SO MUCH!!
    But im stuck on this question now... only b)
    http://www.mathhelpforum.com/math-he...rst-terms.html
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  5. #5
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    "Find the sum of the first ?? terms of the series."

    Something is missing! How many terms?
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  6. #6
    Super Member Bacterius's Avatar
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    Quote Originally Posted by HallsofIvy View Post
    "Find the sum of the first ?? terms of the series."

    Something is missing! How many terms?
    Maybe you have to find out as well ?
    I believe this question intends to make the student apply the formula on the first few terms. But it is indeed ambigious ... how many terms ?
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