Why is the recurrence $f_n=\frac{5f_{n-1} + 1}{25 f_{n-2}}$ cyclic?

Question

Given:

$f_1 = a$

$f_2 = b$

and

$f_n = \dfrac{5f_{n-1} + 1}{25 \cdot f_{n-2}}$

You can just start doing the algebra to show

$f_3 = \dfrac{5b + 1}{25a}$

$f_4 = \dfrac{5a + 5b + 1}{125ab}$

$f_5 = \dfrac{5a + 1}{25b}$

$f_6 = a$

$f_7 = b$ .....

But why is this happening i.e. what in the definition of the recurrence makes this cyclic and determines the cycle length?