\[\large \begin{cases} a_1=2017\\ a_2=2117\\ a_{n+2}=\frac{2}{5}a_{n+1}+\frac{3}{5}a_n, \quad n\ge 1. \end{cases}\]

Let \( \left \{a_n\right \}\) be a sequence satisfying the above condition. Find \(\displaystyle \lim_{n \to \infty } a_n\).

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