Form of Geometric Sequence of Integers/Corollary

Corollary to Form of Geometric Sequence of Integers

Let $p$ and $q$ be integers.

Then the finite sequence $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ defined as:

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

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

Proof

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

Then by Integers Divided by GCD are Coprime:

$p = d r$

$q = d s$

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

Thus:

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

\(\ds a_j\)

\(=\)

\(\ds p^j q^{n - j}\)

\(\ds \)

\(=\)

\(\ds \paren {d r}^j \paren {d s}^{n - j}\)

\(\ds \)

\(=\)

\(\ds d^n r^j s^{n - j}\)

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

Then:

\(\ds \dfrac r s\)

\(=\)

\(\ds \paren {\dfrac p d} / \paren {\dfrac q d}\)

\(\ds \)

\(=\)

\(\ds \dfrac p d \dfrac d q\)

\(\ds \)

\(=\)

\(\ds \dfrac p q\)

Hence the result.

$\blacksquare$