Check that \[ a_{n_2} - a_{n_1} = \frac{a^{n_2}-a^{n_1}}{a-1} =a^{n_1}\cdot \frac{a^{n_2-n_1}-1}{a-1}. \]Take any prime $p\mid n_2-n_1$. Then $p\mid a^p-1$, implying $a$ is coprime to $p$, which in turn implies (via Fermat) that $p\mid a-1$. From here it suffices to observe that for $p>2$, \[ v_p\left(a^{n_2-n_1}-1\right) = v_p(a-1)+v_p(n_2-n_1)\implies v_p\left(\frac{a^{n_2-n_1}-1}{a-1}\right)\ge v_p(n_2-n_1); \]whereas for $p=2$, \[ v_p\left(a^{n_2-n_1}-1\right)-v_p(a-1)\ge v_p(n_2-n_1), \]both of which uses ``lifting the exponent" lemma.