Form of Geometric Sequence of Integers/Corollary
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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.
$\blacksquare$