I am trying to understand the uses of the word "Geometric" throughout mathematics. I suspect that there may be some historical reasons which would tie things together and help my understanding. Supposedly, "Geometric" means "of or relating to Geometry".
"Geometry" is the study of points, lines, surfaces, shapes, etc.
A "Geometric mean" of n numbers is the nth root of the product of the numbers -- how is this related to "Geometry" (shapes, lines, points, etc.)?
A "Geometric series" consists of a sequence of numbers, with each successive number the product of the previous number and some constant -- how does this relate to Geometry?
I know that there have been some attempts by mathematicians to combine Geometry and Algebra into a single framework. Could terms like "Geometric mean" and "Geometric series" have come from these attempts?