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Repeating sequences

  1. Nov 24, 2003 #1
    Is there some concise mathematical form to express the fact that a sequence repeats with period t beginning with the nth term? For example, the sequence {1,2,6,3,7,3,1,7,3,1,7,3,1,7,3,1,...} repeats with period 3 beginning with the 5th term. Can we say, for all n>4, if b=an then b=an+3?
    I need to prove that a certain sequence doesn't repeat. How is this commonly done?
     
  2. jcsd
  3. Nov 24, 2003 #2
    this may work. i'm assuming that the sequence is given by some formula an=f(n). well, if that function is one-to-one, then it won't repeat. if you can prove that f(m)=f(n)-->m=n that would do it. if f is something defined on real numbers and its derivative is never zero, it's one-to-one. that won't exactly help if f has factorials unless you want to get into gamma-land.
     
  4. Nov 24, 2003 #3

    Hurkyl

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    A typical way to state eventual periodicity as:

    [tex]
    \forall n \geq N, a_n = a_{n+e}
    [/tex]
     
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