Form of Geometric Sequence of Integers/Corollary

Corollary to Form of Geometric Progression of Integers
Let $p$ and $q$ be integers.

Then the finite sequence $P = \left\langle{a_j}\right\rangle_{0 \mathop \le j \mathop \le n}$ defined as:


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

is a geometric progression whose common ratio is $\dfrac p q$.

Proof
Let the greatest common divisor of $p$ and $q$ be $d$.

Then by Divide by GCD for Coprime Integers:
 * $p = d r$
 * $q = d s$

where $r$ and $s$ are coprime integers.

Thus:
 * $a_j = p^j q^{n - j}$

and so by Form of Geometric Progression of Integers it follows that $P$ is a geometric progression whose common ratio is $\dfrac r s$.

Then:

Hence the result.