Common Ratio in Rational Geometric Sequence is Rational

Theorem
Let $\left\langle{a_k}\right\rangle$ be a geometric progression whose terms are rational.

Then the common ratio of $\left\langle{a_k}\right\rangle$ is rational.

Proof
Let $r$ be the common ratio of $\left\langle{a_k}\right\rangle$.

Let $p, q$ be consecutive terms of $r$.

By hypothesis $p, q \in \Q$.

Then, by definition of geometric progression:
 * $q = r p$

It follows that:
 * $r = \dfrac q p$

From Rational Numbers form Field, $\Q$ is closed under division.

Thus $r \in \Q$ and hence the result.