Geometric Sequence in Lowest Terms has Coprime Extremes

Proof
Let $a_1, a_2, a_3, \ldots, a_n$ be natural numbers.

Let $\left\langle{a_k}\right\rangle_{1 \mathop \le k \mathop \le n}$ be a geometric progression with common ratio $r$.

Let $a_1, \ldots, a_n$ be the smallest such natural numbers.

From, let $d_1, d_2$ be the smallest natural numbers such that $d_2 = r d_1$.

From one can build a sequence of $3, 4, \ldots, n$ terms with the same property.

Let the geometric progression so constructed with $n$ terms be $\left\langle{b_k}\right\rangle_{1 \mathop \le k \mathop \le n}$.

From, $d_1$ and $d_2$ are coprime.

From, each of the extreme terms of the intermediate geometric progressions is a power of a natural number.

From, each of those extreme terms is coprime.

We have that $a_1, \ldots, a_n$ are the smallest such natural numbers such that $\left\langle{a_k}\right\rangle_{1 \mathop \le k \mathop \le n}$ is a geometric progression with common ratio $r$.

We have also constructed $b_1, \ldots, b_n$ to have the same property.

Thus:
 * $a_1 = b_1, a_2 = b_2, \ldots, a_n = b_n$

and the result follows.