Elements of Geometric Sequence between Coprime Numbers

Theorem
Let $P = \left\langle{a_j}\right\rangle_{0 \mathop \le j \mathop \le n}$ be a geometric progression of integers of length $n + 1$.

Let $a_0$ be coprime to $a_n$.

Then there exist geometric progressions of integers $Q_1$ and $Q_2$ of length $n + 1$ such that:
 * the initial term of both $Q_1$ and $Q_2$ is $1$
 * the final term of $Q_1$ is $a_0$
 * the final term of $Q_2$ is $a_n$.

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

By Form of Geometric Progression of Integers with Coprime Extremes, the $j$th term of $P$ is given by:
 * $a_j = q^j p^{n - j}$

such that:
 * $a_0 = p^n$
 * $a_n = q^n$

For $j \in \left\{{0, 1, \ldots, n}\right\}$, let $r_j = q^j$.

Let the finite sequence $Q_2 = \left\langle{t_j}\right\rangle_{0 \mathop \le j \mathop \le n}$ be defined as:
 * $\forall j \in \left\{{0, 1, \ldots, n}\right\}: t_j = p^{n - j}$

Then $Q_1$ is a geometric progression of integers of length $n$ such that:
 * $t_0 = 1$
 * $t_n = p^n$

Let the finite sequence $Q_1 = \left\langle{s_j}\right\rangle_{0 \mathop \le j \mathop \le n}$ be defined as:
 * $\forall j \in \left\{{0, 1, \ldots, n}\right\}: s_j = q^j$

Then $Q_1$ is a geometric progression of integers of length $n$ such that:
 * $s_0 = 1$
 * $s_n = q^n$

Hence the result.