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How did Euclid’s Elements describe geometric sequences?

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Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

In Euclid’s Elements, geometric sequences are described within the context of number theory rather than plane geometry. Specifically, Book VIII of the Elements examines these sequences, which the Greeks referred to as numbers in "continued proportions".1

Euclid defined these sequences as a series of numbers where the ratio between any term and its predecessor remains constant, represented in modern notation as $a, a x, a x^{2}, a x^{3}, \dots$ . While these sequences are fundamental to modern mathematics, Euclid utilized them to explore properties of integers and to derive specific results, such as his famous "recipe" for identifying perfect numbers. By demonstrating that if the sum of a sequence of powers of 2 (such as $1 + 2 + 4 + \dots + 2^{k}$ ) results in a prime number, then the product of that sum and the last term is a perfect number, Euclid integrated these sequences into the broader logical framework of his work.12