Common Ratio in Rational Geometric Sequence is Rational

Theorem

Let $\sequence {a_k}$ be a geometric sequence whose terms are rational.

Then the common ratio of $\sequence {a_k}$ is rational.

Proof

Let $r$ be the common ratio of $\sequence {a_k}$.

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

We have by hypothesis that:

$p, q \in \Q$

Then, by definition of geometric sequence:

$q = r p$

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.

$\blacksquare$