Powers of Elements of Geometric Sequence are in Geometric Sequence

Theorem

Let $P = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric sequence of integers.

Then the sequence $Q = \sequence {b_j}_{0 \mathop \le j \mathop \le n}$ defined as:

$\forall j \in \set {0, 1, \ldots, n}: b_j = a_j^m$

where $m \in \Z_{>0}$, is a geometric sequence.

In the words of Euclid:

If there be as many numbers as we please in continued proportion, and each by multiplying itself make some number, the products will be proportional; and, if the original numbers by multiplying the products make certain numbers, the latter will also be proportional.

(The Elements: Book $\text{VIII}$: Proposition $13$)

Proof

From Form of Geometric Sequence of Integers, the $j$th term of $P$ is given by:

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

Thus the $j$th term of $Q$ is given by:

$b_j = k^m \paren {q^m}^j \paren {p^m}^{n - j}$

From Form of Geometric Sequence of Integers, this is a geometric sequence.

$\blacksquare$

Historical Note

This proof is Proposition $13$ of Book $\text{VIII}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 2 (2nd ed.) ... (previous) ... (next): Book $\text{VIII}$. Propositions