Form of Geometric Progression of Integers with Coprime Extremes

Theorem

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

Let $a_1$ and $a_n$ be coprime.

Then the $j$th term of $Q_n$ is given by:

$a_j = q^j p^{n - j}$

Proof

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

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

From Geometric Progression with Coprime Extremes is in Lowest Terms, the elements of $Q_n$ are the smallest positive integers such that $Q_n$ has common ratio $r$.

From Form of Geometric Progression of Integers in Lowest Terms the $j$th term of $P$ is given by:

$a_j = q^j p^{n - j}$

where $r = \dfrac p q$.

Hence the result.

$\blacksquare$