Form of Geometric Sequence of Integers in Lowest Terms

Theorem
Let $G_n = \left\langle{a_j}\right\rangle_{0 \mathop \le j \mathop \le n}$ be a geometric progression of length $n$ consisting of positive integers only.

Let $r$ be the common ratio of $G_n$.

Let the elements of $G_n$ be the smallest positive integers such that $G_n$ has common ratio $r$.

Then the $j$th term of $G_n$ is given by:
 * $a_j = p^{n - j} q^j$

where $r = \dfrac q p$.

That is:
 * $G_n = \left({p^n, p^{n - 1} q, p^{n - 2} q^2, \ldots, p q^{n - 1}, q^n}\right)$

Proof
From Form of Geometric Progression of Integers the $j$th term of $G_n$ is given by:
 * $a_j = k p^{n - j} q^j$

where the common ratio is $\dfrac q p$.

Thus:
 * $a_0 = k p^n$
 * $a_n = k q^n$

From Geometric Progression in Lowest Terms has Coprime Extremes it follows that $k = 1$.

Hence the result.