Terminology: geometric sequences and geometric means

Question

(I'll post my own answer to this one, but that should not deter others, since my answer is a surmisal.)

Why are geometric sequences called geometric sequences? Whare are geometric means called geometric means?

Answer 1

Suppose the second of two line segments, $B$, is three times as long as the first $A$, and the third, $C$, is three times as long as the second.

Then the pair $A,B$ has the same geometric shape as the pair $B,C$. That all pairs of consecutive terms in a sequence have the same geometric shape makes it a "geometric sequence".

Answer 2

The arithmetic mean $\mu_a$ of a sequence of numbers $a_1, \dots, a_k$ is the number such that $$ \Sigma a_i = \Sigma \mu_a. $$ If the operation of interest were multiplication rather than addition, the analogous definition would be $$ \Pi a_i = \Pi \mu_g $$ perhaps justifying the use of the term 'mean.' I think this is geometric in the sense that products of lengths are the basis for Euclidean measure, and a cube of side length $\mu_g$ has the same volume as the ($k$-dimensional) rectangle with side lengths $a_1,\dots, a_k$.

Answer 3

Take a square (side length $c$) and a rectangle (side lengths $a$ and $b$) of the same area $A$, then:

$$ A = c^2 = a b \iff c = \sqrt{a b} $$

Otherwise read the Wikipedia articles on arithmetic progression, geometric progression and harmonic progression and look for the remarks on the mean property for each.