Geometric Sequence with Coprime Extremes is in Lowest Terms

Theorem
Let $G_n = \left\langle{a_1, a_2, \ldots, a_n}\right\rangle$ be a geometric progression where $a_1, \ldots, a_n$ are all natural numbers.

Let:
 * $a_1 \perp a_n$

where $\perp$ denotes coprimality.

Then the terms of $G_n$ are the lowest possible natural numbers with the same common ratio.