In this paper, we give the sufficient conditions for the positivity of recurrence sequences defined by $$a_nu_{n}=b_nu_{n-1}-c_nu_{n-2}$$ for $n\geqslant 2$, where $a_n,b_n,c_n$ are all nonnegative and linear in $n$. As applications, we show the positivity of many famous combinatorial sequences.