Construction of Geometric Sequence in Lowest Terms/Proof 2

Theorem
It is possible to find a geometric sequence of integers $G_n$ of length $n + 1$ with a given common ratio such that $G_n$ is in its lowest terms.

Proof
Let the required length of the geometric sequence $P$ be $n$.

Let $r$ be the given common ratio.

From Common Ratio in Integer Geometric Progression is Rational, $r$ is a rational number.

Let $r = \dfrac p q$ be in canonical form.

Thus, by definition:
 * $p \perp q$

Let $a$ be the first term of $P$.

Then the sequence $P$ is:
 * $P = \tuple {a, a \dfrac p q, a \dfrac {p^2} {q^2}, \ldots, a \dfrac {p^n} {q^n} }$

All the elements of $P$ are natural numbers, so, in particular:
 * $a \dfrac {p^n} {q^n} \in \N$

From Powers of Coprime Numbers are Coprime:
 * $p^n \perp q^n$

and so from Euclid's Lemma:
 * $q^n \divides a$

Thus:
 * $a = k q^n$

for some $k \in \N$, and so:
 * $P = \tuple {k q^n, k p q^{n - 1}, k p^2 q^{n - 2}, \ldots, k p^{n - 1} q, k p^n}$

From Geometric Progression with Coprime Extremes is in Lowest Terms:
 * $k q^n \perp k p^n$

from which it follows that $k = 1$.

It follows that the required geometric sequence is:
 * $P = \tuple {q^n, p q^{n - 1}, p^2 q^{n - 2}, \ldots, p^{n - 1} q, p^n}$