Elements of Geometric Sequence from One where First Element is Power 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 $a_1 = 1$.

Let $m \in \Z_{> 0}$.

Let $a_2$ be the $m$th power of an integer.

Then all the terms of $G_n$ are $m$th powers of integers.

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 $a_2 = k^m$.

By definition of geometric progression:
 * $\forall j \in \left\{{2, 3, \ldots, n}\right\}: a_j = r a_{j-1}$

where $r$ is the common ratio.

This holds specifically for $j = 2$:
 * $k^m = r \cdot 1$

Thus:

Hence the result.