Product of Geometric Sequences from One

Theorem
Let $Q_1 = \left\langle{a_j}\right\rangle_{1 \mathop \le j \mathop \le n}$ and $Q_2 = \left\langle{b_j}\right\rangle_{1 \mathop \le j \mathop \le n}$ be geometric progressions of integers of length $n$.

Let $a_1 = b_1 = 1$.

Then the sequence $P = \left\langle{p_j}\right\rangle_{1 \mathop \le j \mathop \le n}$ defined as:
 * $\forall j \in \left\{{1, \ldots, n}\right\}: p_j = b^{j - 1} a^{n - j}$

is a geometric progression.

Proof
By Form of Geometric Progression of Integers with Coprime Extremes, the $j$th term of $Q_1$ is given by:
 * $a_j = a^{j - 1}$

and of $Q_1$ is given by:
 * $b_j = b^{j - 1}$

Let the terms of $P$ be defined as:
 * $\forall j \in \left\{{1, \ldots, n}\right\}: p_j = b^{j - 1} a^{n - j}$

Then from Form of Geometric Progression of Integers it follows that $P$ is a geometric progression.