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Two sequences defined for all naturals by

  1. Jan 13, 2005 #1

    quasar987

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    Consider [itex]x_1,y_1 \in \mathbb{R}[/itex] such that [itex]x_1>y_1>0[/itex] and [itex]\{x_n\},\{y_n\}[/itex] the two sequences defined for all naturals by

    [tex]x_{n+1}=\frac{x_n+y_n}{2}, \ \ \ \ \ y_{n+1}=\sqrt{x_n y_n}[/tex]

    Show that the sequence [itex]\{y_n\}[/itex] is increasing and as [itex]x_1[/itex] for an upper bound.


    I would appreciate some help on this one, I have made no progress.
     
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  3. Jan 13, 2005 #2

    mathman

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    The key to the proof is to show xn+1>yn+1. Since they are both positive, this is equivalent to showing the squares exhibit the same inequality. A simple calculation will show this to be true.

    xn+12-yn+12=((xn-yn)/2)2.

    The rest is easy. xn>xn+1> ,yn<yn+1.

    Therefore your desired conclusions hold.
     
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