# Where does the name "geometric sequence" come from?

On this and other Stack Exchange website, there have been question about the so-called geometric series, and where its name comes from. My problem is that most answers follow one of two different ideas.

1. Either they show an ingenious, but obviously not ancient, picture that illustrates concrete examples of geometric series summation.
2. Or they make some reference to the middle of three consecutive terms in the sequence being the "geometric mean" of the other two.

Well, I want to raise the question again, because neither of these prototype answers really explains where the use of "geometric" started, nor who (if known) came up with it.

Any good explanation?

• The "why" is not clear. The origin is with the Pythagorean School (see also: Archytas. The early extant souce seems to be Fragment 2 of the lost work of OnMusic of Archytas. Jul 22 '17 at 8:23
• @Mauro Fragment 2 seems to be the closest we can be to an answer! Wow! If you migrate your comment to an answer, I will gladly accept it. Jul 23 '17 at 2:56
• Although the phrases "arithmetic mean", "geometric mean", and "harmonic mean" have a history that go back to the Greeks, is it really a stretch that 16th and 17th century mathematicians (who knew their classical mathematics) decided to refer to $\sum(a+kb)$, $\sum x^k$, $\sum 1/k$ as "arithmetic series", "geometric series", and "harmonic series" respectively since the value between any two terms is that particular mean?
– user6918
Jan 16 '18 at 15:33
• I can even imagine a history where the originator of the terms thought it was a natural thing to call them and only had to take 2 seconds to explain the nomenclature to an acquaintance; everyone immediately understood and employed the terms without ever thinking of writing down the explanation so that future generations could have no doubt behind their motivations.
– user6918
Jan 16 '18 at 15:36

The "why" is not clear.