How to prove that the Fibonacci sequence is periodic mod 5 without using induction? The sequence $(F_{n})$ of Fibonacci numbers is defined by the recurrence relation
$$F_{n}=F_{n-1}+F_{n-2}$$ 
for all $n \geq 2$
with $F_{0} := 0$
and 
$F_{1} :=1$.
Without mathematical induction,
how can I show that
$$F_{n}\equiv F_{n+20}\pmod 5$$
for all $n \geq 2$?

(This is @mathlove 's solution slightly streamlined.)
The sequence 
$$G_n:=F_{n+5}-3F_n\qquad({\rm mod} \ 5)$$
satisfies the same recursion as the $F_n$. Furthermore one easily checks that $G_0=G_1=0$. This implies $G_n=0$ for all $n$, so that $$F_{n+5}=3F_n \qquad({\rm mod} \ 5)$$
for all $n$. It follows that $$F_{n+20}=3^4 F_n =F_n\qquad({\rm mod} \ 5)$$
for all $n$.
