Limit of a Recursively Defined Sequence

Let $(a_1,a_2,a_3\dots)$ be an infinite sequence of real numbers such that for all positive integers $n$, $a_1 = 1$ and $a_{n+1} = \sqrt{\frac{20a_n+7}{3}}$. Evaluate $\displaystyle \lim_{n\to\infty} a_n$.