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Sequences / Real Analyses question

  1. Apr 19, 2009 #1
    Sequences / Real Analysis question

    1. The problem statement, all variables and given/known data
    a,b are the roots of the quadratic equation x2 - x + k = 0, where 0 < k < 1/4.
    (Suppose a is the smaller root). Let h belong to (a,b). The sequence xn is defined by:
    [tex]x_1 = h, x_{n+1} = x^2_n + k. [/tex]

    Prove that a < xn+1 < xn < b, and then determine the limit of xn.


    2. Relevant equations



    3. The attempt at a solution
    I have no idea how to start, if you could help me.
    Thanks.
     
    Last edited: Apr 19, 2009
  2. jcsd
  3. Apr 19, 2009 #2

    mjsd

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    Homework Helper

    perhaps starts by determining a and b in terms of k?
     
  4. Apr 19, 2009 #3
    Okay, so I got [tex]a = \frac{1 - \sqrt{1 - 4k}}{2}, b = \frac{1 + \sqrt{1 - 4k}}{2}[/tex].

    And I was able to prove [tex]X_{n+1} < X_n [/tex] by induction. But, I'm stuck on the outer inequalities.

    EDIT: [tex]X_{n+1} < X_n [/tex] means that X1 = h is the largest value of Xn for all n. And h belongs to (a,b), so X1 < b, and consequently Xn < b.

    I still need to prove that a is a lower bound..
     
    Last edited: Apr 19, 2009
  5. Apr 19, 2009 #4
    I think the basic idea is:
    [tex]X_{n+1} < X_n \Leftrightarrow X^2_n - X_n + k < 0[/tex]

    Therefore, Xn must be between the roots for this equation to be negative.
    But is there a more mathematical way to state it?
     
  6. Apr 19, 2009 #5
    Good job! How about saying x^2-x+k=(x-a)(x-b) which is negative if and only if a<x<b.
     
  7. Apr 19, 2009 #6
    Oh right! Thanks a lot :)
     
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