Limit of a two-term average recurrence

Let $\{a_{n}\}_{n\ge 1}$ be a sequence of real numbers such that $a_{1}=1$, $a_{2}=7$ and $a_{n+1}=\dfrac{a_{n}+a_{n-1}}{2}$, $n\ge 2$. Assuming that $\lim_{n\to\infty}a_{n}$ exists, the value of $\lim_{n\to\infty}a_{n}$ is