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Recursive sequence problem?

  1. Apr 14, 2014 #1
    1. The problem statement, all variables and given/known data

    Let [itex]\{P_i\}_{i=0}^\infty [/itex] be a sequence of points on a plane. Suppose [itex]P_i[/itex]s are placed as on the picture below, so that [itex] |P_0 P_1|=2, |P_1 P_2|=1, |P_2 P_3|=.5, |P_3P_4|=.25[/itex], ... Find the coordinate of the point [itex]P = \lim_{i→\infty} P_i [/itex]

    attachment.php?attachmentid=68646&d=1397459162.png

    2. Relevant equations



    3. The attempt at a solution

    here are the points


    [itex] P_0: (0,0) P_1: (2,0) P_2: (2, 1) P_3: (1.5, 1) P_4: (1.5, .75) P_5: (1.625, .75) [/itex]

    lets examine the x values first:
    2, 1.5, 1.625

    this is a sequence defined recursively by:

    [itex] a_1 = 2 [/itex]

    [itex] a_{n+1} = 2 - \frac{a_n}{4} [/itex]

    [itex] L = \lim_{a_n\rightarrow\infty} a_n = \lim_{a_n\rightarrow\infty} a_{n+1}
    = \lim_{an\rightarrow\infty} 2-\frac{a_n}{4} [/itex]

    which means that

    [itex] L = 2-\frac{L}{4} [/itex]

    [itex] 4L = 8 - L [/itex]

    [itex] 5L = 8 [/itex]

    [itex] L = \frac{8}{5} [/itex]

    [itex] L = 1.6 [/itex]

    so 1.6 would be the x coordinate of the point.

    I then would follow a similar process to find the y coordinate, but before I do that I just wanna make sure what I have so far is correct.
     

    Attached Files:

  2. jcsd
  3. Apr 14, 2014 #2
    also forgive me for not proving the sequence is bounded and decreasing (I left it out to save time and narrow the focus) but i have done so by mathematical induction
     
  4. Apr 14, 2014 #3
    Yes it is correct but also prove it is bounded and decreasing.
     
  5. Apr 15, 2014 #4
    Ok so it turns out one of my induction proofs was wrong. I can prove it is bounded but this sequence is not strictly increasing or decreasing. it hops back and forth between increasing and decreasing but it IS approaching a certain point. Is there any way to prove that it has a limit?
     
  6. Apr 15, 2014 #5
    the proof that it is bounded:

    [itex] 1 ≤ a_n ≤ 2 [/itex]

    [itex] \frac{1}{4} ≤ \frac{a_n}{4} ≤ \frac{2}{4} [/itex]

    [itex] -\frac{1}{4} ≥ -\frac{a_n}{4} ≥ -\frac{2}{4} [/itex]

    [itex] 2-\frac{1}{4} ≥ 2-\frac{a_n}{4} ≥ 2-\frac{2}{4} [/itex]

    [itex] 1.75 ≥ a_{n+1} ≥ 1.5 [/itex]

    [itex] 1 ≤ 1.5 ≤ a_{n+1} ≤ 1.75 ≤ 2 [/itex]

    [itex] 1 ≤ a_{n+1} ≤ 2 [/itex]
     
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