Hyers-Ulam stability of the first order difference equation generated by linear maps

Hyers-Ulam stability of the difference equation $z_{n+1}=a_n{z}_n+b_n$ is investigated. If ${\prod }_{j=1}^n\mathrm{\mid }{a}_j\mathrm{\mid }$ has subexponential growth rate, then difference equation generated by linear maps has no Hyers-Ulam stability. Other complementary results are also found where $ \lim_{n \rightarrow \infty} \left(\prod_{j=1}^{n}|a_j| \right)^{\frac{1}{n}} $ is greater or less than one. These results contain Hyers-Ulam stability of the first order linear difference equation with periodic coefficients also.