AIME 2009a Problem 7

The sequence $(a_n)$ satisfies $a_1 = 1$ and $$5^{(a_{n + 1} - a_n)} - 1 = \frac{1}{n + \frac{2}{3}}$$ for $n \geq 1$. Let $k$ be the least integer greater than $1$ for which $a_k$ is an integer. Find $k$.