If First of Three Numbers in Geometric Sequence is Square then Third is Square

Theorem
Let $P = \tuple {a, b, c}$ be a geometric sequence of integers.

Let $a$ be a square number.

Then $c$ is also a square number.

Proof
From Form of Geometric Sequence of Integers:
 * $P = \tuple {k p^2, k p q, k q^2}$

for some $k, p, q \in \Z$.

If $a = k p^2$ is a square number it follows that $k$ is a square number: $k = r^2$, say.

So:
 * $P = \tuple {r^2 p^2, r^2 p q, r^2 q^2}$

and so $c = r^2 q^2 = \paren {r q}^2$.