The recurrence an=0.8an−1+0.15an−2a_n = 0.8a_{n-1} + 0.15a_{n-2}an=0.8an−1+0.15an−2 converges to 0 as n→∞n \to \inftyn→∞ because:
Both characteristic roots have modulus less than 1
The recurrence is homogeneous
The sum of coefficients is less than 1
The initial conditions are positive