MathNet 2013 Question 10

Let $a$ and $n$ be integers. We define $a_n = 1 + a + a^2 + \dots + a^{n-1}$. Prove that if $a^p \equiv 1 \pmod p$ for every prime divisor $p$ of $n_2 - n_1$, then the number $\frac{a_{n_2} - a_{n_1}}{n_2 - n_1}$ is an integer.