Elements of Geometric Sequence from One which are Powers of Number

Theorem
Let $G_n = \left\langle{a_n}\right\rangle_{1 \mathop \le i \mathop \le n}$ be a geometric progression of integers.

Let $a_1 = 1$.

Then:
 * $\forall m, k \in \left\{{2, \ldots, n}\right\}: k \mathop \backslash \left({m - 1}\right) \implies a_m$ is a power of $k$

where $\backslash$ denotes divisibility.

Proof
By Form of Geometric Progression of Integers from One, the general term of $G_n$ can be expressed as:
 * $a_j = q^{j - 1}$

for some $q \in \Z$.

Let $k, m \in \left\{{2, \ldots, n}\right\}$ such that $k \mathop \backslash \left({m - 1}\right)$.

By definition of divisibility:
 * $\exists r \in \Z: \left({m - 1}\right) = r k$

Then:

That is, $a_m$ is a power of $k$.