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Sequences and Series Problem

  1. Apr 24, 2013 #1
    1. The problem statement, all variables and given/known data

    The sum of the first 9 terms of an arithmetic series is 216. The first, third and seventh terms of the series form the first three terms of a Geometric pattern. Find the first term and the common difference of the arithmetic pattern.

    2. Relevant equations



    3. The attempt at a solution

    S9 = 9/2(2a+8d) = 216
    S3 = T1+T3+T7

    Completely stuck and need help
     
    Last edited by a moderator: Apr 24, 2013
  2. jcsd
  3. Apr 24, 2013 #2

    mfb

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    What are a and d?
    Can you express T1, T3 and T7 as functions of a and d? Do you know what "geometric pattern" means?
     
  4. Apr 24, 2013 #3
    Okay,
    T1 = a
    T3 = a+2d
    T7 = a+6d

    A geometric pattern is one with a constant ratio
     
  5. Apr 24, 2013 #4

    mfb

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    Don't just say it, express it as formula :).
     
  6. Apr 24, 2013 #5
    Tn = ar^n-1
     
  7. Apr 24, 2013 #6

    mfb

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    Oh come on, you can try more than one step per post...
     
  8. Apr 24, 2013 #7

    vela

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    This isn't correct. The third partial sum, S3, is always T1+T2+T3.

    The problem statement didn't say anything about S3. It just said T1, T3, and T7 are the first three terms of a geometric progression.
     
  9. Apr 24, 2013 #8
    So how would I bring the geometric pattern into all of this?
     
  10. Apr 24, 2013 #9

    vela

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    That's what you're supposed to figure out. :smile:

    You should use different letters here for the two different sequences. You've used Tn to denote the terms in the arithmetic sequence, and Sn to denote the partial sums formed from Tn. So to summarize what you have so far:
    \begin{align*}
    T_n &= a+(n-1)d \\
    S_9 &= T_1 + T_2 + \cdots + T_9 = \frac{9}{2}(2a+8d)
    \end{align*} You also said a geometric sequence has terms of the form ##G_n = br^{n-1}##. I used G instead of T because you have two different sequences, and I used ##b## instead of ##a## because you already used the letter ##a## as one of the parameters for the arithmetic sequence.

    In terms of the T's and G's, how would you express the sentence "The first, third and seventh terms of the (arithmetic) series form the first three terms of a geometric pattern"?
     
  11. Apr 25, 2013 #10
    (a)+(a+2d)+(a+6d) = (a)+(ar)+(ar^2)
     
  12. Apr 25, 2013 #11

    mfb

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    Why do you want to add anything? Each term of your series should correspond to one term of the geometric series. This gives 3 equations without a sum, not one.
    As vela pointed out, please do not use "a" for two different things. Use b instead for the geometric series. In addition, it could help if you follow vela's advice and use Tn, Gn first for those equations. You can plug in the known formulas for them afterwards.
     
  13. Apr 25, 2013 #12
    Ok so,
    a = b
    a+2d = br
    a+6d = br^2
     
  14. Apr 25, 2013 #13

    vela

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    Looks good. Keep going.
     
  15. Apr 25, 2013 #14
    Since they are in geometric series. Then (a+7d)/(a+2d) = (a+2d)/(a)
     
  16. Apr 25, 2013 #15
    From the first equation about the sum of the 9 terms u can write a equation in terms of a and d , then use system of equation to solve for a and d
     
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