Q:

What is the history of the meanings behind the word "Geometric"?

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?

A:

"Geometry" literally translates from Greek as "land measurement", and reflected a popular Greek belief that they learned it from Egyptian "rope stretchers", land meausurers, who used ropes to perform what came to be called straightedge and compass constructions. 
Geometric mean for $n=2$, or mean proportional as Greeks called it, was likely so named because it answers a geometric question: what is the side of a square having the same area as a rectangle with given sides. Euclid gives a straightedge and compass construction for it in Elements VI.13, but it was already known to early Pythagoreans in 6th century BC.
Geometric series was likely so named because each of its terms is the geometric mean of its immediate neighbors $a_n=\sqrt{a_{n-1}a_{n+1}}$ (assuming all terms are positive). The same is true of arithmetic and harmonic series, their terms are also the corresponding means of their neighbors. The notion if not the name is also attributed to early Pythagoreans, Euclid expresses in words what we call partial sum of the geometric series in Elements IX.35.

A:

Geometric mean can be understood in terms of geometry. The geometric mean (or mean proportional) of two numbers, $a$ and $b$, is the length of the side of a square whose area is equal to the area of a rectangle with sides of lengths $a$ and $b$. 
In other terms, is a number $c$ such that :

$a \times b = c \times c$

that comes from :

$$\frac a c = \frac c b.$$

See Euclid's Elements VI.13.

