What are geometric sequences in Book VIII of Euclid’s Elements?

In Book VIII of the Elements, Euclid examines numbers in continued proportions, which are now known as geometric sequences.1

These sequences are represented by terms such as $a·x$, $a·x^2$, $a·x^3$, $a·x^4$, and so on. While the Elements is often associated primarily with geometry, Books VII through IX focus on number theory, where Euclid defines numbers as collections of units (positive integers greater than 1). Book VIII specifically develops propositions concerning these numerical progressions, as well as square and cube numbers.12