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Math Help - comparison between sum of two sequence

  1. #1
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    comparison between sum of two sequence

    in an arithmetic progression and geometric progression the total number of terms are same.the first term and last term of both progressions are also same. if A and G are sum of these expressions then what is relation between the sums.
    i tried every way but not able to find rigid proof ?
    Last edited by ayushdadhwal; December 16th 2013 at 08:32 AM.
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  2. #2
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    Re: comparison between sum of two sequence

    I will help you out by providing some notation. The arithmetic progression can be represented by the terms a_0, \ldots, a_n and the geometric progression can be represented by the terms g_0, \ldots, g_n. For a progression to be arithmetic, there must be some real number d such that a_k = a_0 + kd. For a progression to be geometric, there must be some real number r with g_k = g_0r^k. Next, let's figure out the arithmetic and geometric sums:

    A = \sum_{k=0}^n a_k = \sum_{k=0}^n (a_0 + kd) = a_0\sum_{k=0}^n 1 + d\sum_{k=0}^n k

    G = \sum_{k=0}^n g_k = \sum_{k=0}^n g_0r^k = g_0\sum_{k=0}^n r^k
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  3. #3
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    Re: comparison between sum of two sequence

    Quote Originally Posted by ayushdadhwal View Post
    [TD="class: post-message, bgcolor: #F9F9F9"]in an arithmetic progression and geometric progression the total number of terms are same.the first term and last term of both progressions are also same. if A and G are sum of these expressions then what is relation between the sums.[/TD][/TR][TR="class: even, bgcolor: #F9F9F9"][/TR][/TABLE]i tried every way but not able to find rigid proof ?
    Quote Originally Posted by SlipEternal View Post
    I will help you out by providing some notation. The arithmetic progression can be represented by the terms a_0, \ldots, a_n and the geometric progression can be represented by the terms g_0, \ldots, g_n. For a progression to be arithmetic, there must be some real number d such that a_k = a_0 + kd. For a progression to be geometric, there must be some real number r with g_k = g_0r^k. Next, let's figure out the arithmetic and geometric sums:

    A = \sum_{k=0}^n a_k = \sum_{k=0}^n (a_0 + kd) = a_0\sum_{k=0}^n 1 + d\sum_{k=0}^n k

    G = \sum_{k=0}^n g_k = \sum_{k=0}^n g_0r^k = g_0\sum_{k=0}^n r^k
    In reply #2 you must realize that the number of terms is n+1.

    I would be the first to say that I have no idea what the author of this question expects as an answer.
    Because both sequences have the same first term, lets call it a.

    Then the last term being equal we get [TEX]a+nd-=ar^n [TEX]
    Last edited by Plato; December 16th 2013 at 04:23 PM.
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    Re: comparison between sum of two sequence

    Quote Originally Posted by Plato View Post
    In reply #2 you must realize that the number of terms is n+1.

    I would be the first to say that I have no idea what the author of this question expects as an answer.
    Because both sequences have the same first term, lets call it a.

    Then the last term being equal we get [TEX]a+nd-=ar^n [TEX]
    Sir we have to prove that sum A is greater than or equal to G
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  5. #5
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    Re: comparison between sum of two sequence

    Solve for d in terms of a, n and r. Then evaluate the sums.
    Last edited by SlipEternal; December 16th 2013 at 09:22 PM.
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    Re: comparison between sum of two sequence

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