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Homework Help: Sequence and Series, finding relationship

  1. Nov 25, 2007 #1
    1. The problem statement, all variables and given/known data

    0<y0<x0
    x1=(x0+y0)/2
    y1=[tex]\sqrt{x0y0}[/tex]

    in general
    Xn+1=(xn+yn)/2
    Yn+1=[tex]\sqrt{XnYn}[/tex]

    2. Relevant equations

    none

    3. The attempt at a solution

    I have no idea
    I tried to solve for Xn and substituting that into another equation...
    however, I don't know how to simplify it down to one single variable....

    How do I take the limit of the sequence, so that I can find the relationship
     
  2. jcsd
  3. Nov 25, 2007 #2

    EnumaElish

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    Please state the problem as it is given.
     
  4. Nov 25, 2007 #3
    the problem was given that way, we are trying to find the relationship between X and y
     
  5. Nov 25, 2007 #4

    rock.freak667

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    take the general equations and sub one into the other is all i can suggest
     
  6. Nov 26, 2007 #5

    HallsofIvy

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    Since there was NO X or Y in what you gave, I don't see how that can be true.
     
  7. Nov 26, 2007 #6

    VietDao29

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    Hmm, I think I got what you mean.

    Given 0 < y0 < x0, the sequence (xn), and (yn) are defined as follow:

    [tex]x_{n + 1} = \frac{x_n + y_n}{2}, n \in \mathbb{N}[/tex] (1)
    [tex]y_{n + 1} = \sqrt{x_n y_n}, n \in \mathbb{N}[/tex] (2)

    Now find the limit of xn, and yn (Or does it tell you to prove that the limit of the two sequences are the same?).

    Is that the correct problem?

    ---------------------------

    Ok, now you must at least have a vision of how xn, and yn behave. Draw a pictures like this:

    0____y0____________________________x0

    Now, we have: x1 = (x0 + y0) / 2, that means, x1 lies exactly at the middle of x0, and y0.

    [tex]y_1 = \sqrt{x_0 y_0} > \sqrt{y_0 ^ 2} = y_0[/tex]

    You'll also have: [tex]x_1 - y_1 = \frac{x_0 - 2\sqrt{x_0 y_0} + y_0}{2} = \frac{(\sqrt{x_0} - \sqrt{y_0}) ^ 2}{2} > 0[/tex], so x1 > y1.

    0____y0____y1_________x1_______________x0

    Now, where do x2, and y2 lie?

    ---------------------------

    So, in conclusion, let's answer some questions:

    1. xn, and yn, which is greater?

    2. Is (xn) an increase sequence, or a decrease sequence? Can you prove it?

    3. Is (yn) an increase sequence, or a decrease sequence? Can you prove it?

    4. Are they bounded? Do they have limit?

    5. Do they both have the same limit?

    Ok, I think you can take it from here. Can you? :)
     
    Last edited: Nov 26, 2007
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