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Finding a Formula for the General Term of a Sequence

  1. Jul 31, 2013 #1
    1. The problem statement, all variables and given/known data
    A bored student enters the number 0.5 in her calculator, then repeatedly computes the square of the number in the display. Taking A0 = 0.5, find a formula for the general term of the sequence {An} of the numbers that appear in the display, and find the limit of the sequence {An}.


    2. Relevant equations



    3. The attempt at a solution

    So, I'm having a difficult time finding a standard rule to this sequence. The only thing I came up with was an= (a(n-1))2. Then it doesn't seem reasonable to show the limit of this sequence is = 0 as n increases, an decreases. Is there another way of finding a rule that is easier to work with?
     
  2. jcsd
  3. Jul 31, 2013 #2

    micromass

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    So ##a_0 = 0.5##. Can you give me ##a_1##? ##a_2##? ##a_3##? ##a_4##? (don't work it out completely, leave it in symbols, so don't say that ##(0.5)^2 = 0.25##, but leave it as ##(0.5)^2##). Do you notice a pattern?
     
  4. Jul 31, 2013 #3
    ##a_1##=##(0.5)^2##

    ##a_2##=##(0.25)^2##

    ##a_3##=##(0.0625)^2##

    ##a_4##=##(0.00390625)^2##

    Hmmm, I see that 0625 is recurring and I'm assuming that as n increases, the amount of decimal places increase by 2n places. Does that make sense?

    EDIT: This is probably what you didn't want me to do eh Micromass? I improved my answer down below.
     
    Last edited: Jul 31, 2013
  5. Jul 31, 2013 #4

    LCKurtz

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    You are simplifying, which hides the pattern. And so do decimals. Start out with ##a_0=\frac 1 2## and try that. Look for un-multiplied out powers of ##2##.
     
  6. Jul 31, 2013 #5
    Gotcha,

    ##a_0=\frac 1 {2}^{1}##
    ##a_1=\frac 1 {2}^{2}##
    ##a_2=\frac 1 {2}^{4}##
    ##a_3=\frac 1 {2}^{8}##

    Bingo. ##a_n={2}^{2}^{n}##

    I can't seem to get latex to work but it's 22n

    EDIT: I got excited, it's 1/22n
     
  7. Jul 31, 2013 #6

    LCKurtz

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    I added parentheses which are necessary and fixed your latex. Right click on an expression to see how it was fixed.
     
  8. Jul 31, 2013 #7
    Thank you so much LCKurtz
     
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