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What is the general term of this sequence?

  1. Apr 20, 2013 #1
    Could you help me to find the general term of the sequence:

    ## 1 , \frac{5}{3} , 1 , \frac{15}{17} , 1 , \frac{37}{35} , 1 , \frac{63}{65} ,... ##

    Thank you!
     
  2. jcsd
  3. Apr 20, 2013 #2

    mfb

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    There are infinitely many general expressions for series which begin like that.
    If you look for the "easiest" expression, there could be something simple, but I don't see it at the moment.
    The numerator seems to follow the pattern (same as denominator, 2 more, same, 2 less, same, 2 more, ...).
    I don't see a clear pattern for the denominator, however.
     
  4. Apr 20, 2013 #3

    micromass

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    Think about perfect squares.
     
  5. Apr 20, 2013 #4
    Perfect squares? How? I don't see it
     
  6. Apr 20, 2013 #5

    MarneMath

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    Well, 5/3 is (2^2 + 1)/(2^2 - 1) and 15/17 = (4^2 - 1)/(4^2+1) and 37/35 = ( 6^2 +1)/(6^2-1) and so on and so forth.
     
  7. Apr 21, 2013 #6

    mfb

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    Oh, nice. I was too focused on powers of 2, which are close (+1, or +3 in one case) to all of the "visible" denominators.
     
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