Form of Geometric Sequence of Integers with Coprime Extremes
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Theorem
Let $Q_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence 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 Sequence 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 Sequence 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.
$\blacksquare$