# Repeating sequences

1. Nov 24, 2003

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

2. Nov 24, 2003

### phoenixthoth

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.

3. Nov 24, 2003

### Hurkyl

Staff Emeritus
A typical way to state eventual periodicity as:

$$\forall n \geq N, a_n = a_{n+e}$$