Proving a Sequence Doesn't Repeat: What's the Best Method?

[StephenPrivitera]

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=a_n then b=a_n+3?
I need to prove that a certain sequence doesn't repeat. How is this commonly done?

 

[phoenixthoth]

this may work. I'm assuming that the sequence is given by some formula a_n=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.

 

[Hurkyl]

A typical way to state eventual periodicity as:

<br /> \forall n \geq N, a_n = a_{n+e}<br />