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Thread: Show the sequence is decreasing

  1. #1
    Junior Member universalsandbox's Avatar
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    Show the sequence is decreasing

    I have attached a picture of the question. How would you go about showing this? Thanks.

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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by universalsandbox View Post
    I have attached a picture of the question. How would you go about showing this? Thanks.

    Is/are there an extra condition/s that is/are missing here?

    RonL
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  3. #3
    Junior Member universalsandbox's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    Is/are there an extra condition/s that is/are missing here?

    RonL
    Ex:

    let a=2, xknot = 1

    then

    x(sub1) = (1/2)(xknot + 2/xknot) = 3/2

    x(sub2) = (1/2)( x(sub1) + 2/(xsub1) ) = 17/2 etc.
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  4. #4
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    I work out that if $\displaystyle x_n > \frac{a}{x_n}$,

    $\displaystyle x_n + \frac{a}{x_n} > \frac{2a}{x_n}$

    $\displaystyle 0.5\left(x_n + \frac{a}{x_n}\right) > \frac{a}{x_n}$

    $\displaystyle x_{n+1} > \frac{a}{x_n}$.

    As long as $\displaystyle x_n > \frac{a}{x_n}$, then the function will be decreasing, but I can't find a way to show that statement inductively.
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by universalsandbox View Post
    Ex:

    let a=2, xknot = 1

    then

    x(sub1) = (1/2)(xknot + 2/xknot) = 3/2

    x(sub2) = (1/2)( x(sub1) + 2/(xsub1) ) = 17/2 etc.
    Compare what happens when $\displaystyle x_0>\sqrt{a}$ with what happens when $\displaystyle x_0<\sqrt{a}$ with what happens when $\displaystyle x_0=\sqrt{a}.$

    RonL
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