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.

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.