Form of Geometric Sequence of Integers in Lowest Terms

Theorem
Let $P = \left\langle{a_j}\right\rangle_{1 \mathop \le j \mathop \le n}$ be a geometric progression of length $n$ consisting of positive integers only.

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

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

Then the $j$th term of $P$ is given by:
 * $a_j = q^{j - 1} p^{n - j}$

where $r = \dfrac p q$.

Proof
From Form of Geometric Progression of Integers the $j$th term of $P$ is given by:
 * $a_j = k q^{j - 1} p^{n - j}$

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

Thus:
 * $a_1 = k q^{n - 1}$
 * $a_n = k p^{n - 1}$

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

Hence the result.