How can we be sure of periodicity by testing some terms? A mod $n$ Fibonacci sequence is simply defined as the Fibonacci sequence, except all terms are in mod $n$.
Now to determine periodicity, the worked solutions computed the first 20 or so terms and then observed that it was periodic with period 16.
My question is: Just by testing terms like that, how do we REALLY know for sure that the period is 16?
How do we know that say when we test the first 20 terms, the 17th, 18th, 19th etc match the 2nd, 3rd, 4th etc term respectively, but then everything else goes wrong afterwards?

If $F_{16} \equiv F_0 \equiv 0 \text{ modulo }n$ and  $F_{17} \equiv F_1 \equiv 1 \text{ modulo }n$ then you can use induction to show $F_{16+k} \equiv F_k \text{ modulo }n$ for all non-negative integer $k$.
This does not guarantee that the period is $16$, but it does ensure the period divides $16$: for example with $n=3$ you have $F_{16+k} \equiv F_k \text{ modulo }3$ but the period is $8$.  So to ensure the period is exactly $16$ you also need to check it is not shorter, i.e. that it is not $1,2,4,8$. 
