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Math Help - Question on finite series?

  1. #1
    Super Member fardeen_gen's Avatar
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    Question on finite series?

    Find all sequences which are simultaneously an Arithmetic Progression and a Geometric Progression.
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  2. #2
    MHF Contributor kalagota's Avatar
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    a_n = a_1 + (n-1)d --- Arithmetic prog.
    a_n = a_1r^{n-1} --- Geometric prog.

    from geometric prog.. we have a_1 = \frac{a_n}{r^{n-1}}

    so, we have a_n = \frac{a_n}{r^{n-1}} + (n-1)d \Longleftrightarrow a_nr^{n-1} = a_n + (n-1)dr^{n-1} \Longleftrightarrow a_n = \frac{(n-1)dr^{n-1}}{r^{n-1}-1}
    Last edited by kalagota; July 14th 2008 at 06:53 AM.
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  3. #3
    Super Member fardeen_gen's Avatar
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    --- Arithmetic prog.
    I think for an Arithmetic Progression,
    a(subscript n) = a(subscript 1) + (n - 1)d, isn't it?
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  4. #4
    MHF Contributor kalagota's Avatar
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    oh yes.. just a typo error..
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  5. #5
    Super Member flyingsquirrel's Avatar
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    Hi
    Quote Originally Posted by fardeen_gen View Post
    Find all sequences which are simultaneously an Arithmetic Progression and a Geometric Progression.
    Let (a_n)_{n\geq 0} be such a sequence. There exists two real numbers d and r such that for any non-negative integer n

    \begin{cases} a_{n+1}-a_n=d & (1)\\ a_{n+1}=ra_n & (2)\end{cases}


    Substitute (2) in (1) : ra_n-a_n=d \,\,\,(3)

    • If r\neq 1 , (3)\implies  a_n=\frac{d}{r-1} hence (a_n) is a constant sequence. This implies that the common difference has to be 0 hence a_n=\frac{0}{r-1}=0 for any non-negative integer n.
    • If r=1 , (3) \implies d=a_n-a_n=0 hence a_n=a_0 for any non-negative integer n and (a_n) is a constant sequence.


    At this point we've shown that if a progression is both arithmetic and geometric than the sequence is constant. Now, one has to show that if a sequence is constant then it is both arithmetic and geometric : it will allow us to claim that "A sequence is both arithmetic and geometric iff it is constant".
    Last edited by flyingsquirrel; July 14th 2008 at 08:43 AM. Reason: integer => non-negative integer
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