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Just curious about why geometric progression is called so. Is it related to geometry?

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Here's an extremely nice picture illustrating the geometric series and making it geometrically clear how it converges when $x<1$.

Dobbs 1918

$s-1=xs$ follows from $PN/ON=BA/OA$ which follows from the fact that $\triangle OAB$ is similar to $\triangle ONP$.

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    $\begingroup$ Great added bonus for the (unfamiliar to me) proof for the sum :) $\endgroup$ May 15, 2015 at 4:11
  • $\begingroup$ @nbubis Right, I only knew how to make the geometric series intuitive and visual for $x=1/2$, but this allows one to do it for all $x\in(0,1)$. $\endgroup$ May 15, 2015 at 17:47
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    $\begingroup$ What book is this? $\endgroup$
    – Alec Teal
    May 16, 2015 at 18:13
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    $\begingroup$ @AlecTeal It's this one: amazon.com/Plane-Trigonometry-H-S-Carslaw/dp/B00BGGHADM/… $\endgroup$ Jun 7, 2015 at 12:51
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    $\begingroup$ @AlecTeal The image is not from a book; it comes from page 245 of this paper: W.J. Dobbs, "The Introduction to Infinite Series", The Mathematical Gazette 9, no. 135 (May 1918), pp. 242--246, DOI: 10.2307/3605310. $\endgroup$
    – Senex
    Feb 14, 2019 at 14:18
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Here is a geometric figure illustrating the geometric progression $1,r,r^2,r^3,r^4,r^5,\ldots$:

enter image description here

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    $\begingroup$ Now that's a good picture! We don't have to worry about whether people were thinking about $8$-dimensional cubes, however long ago the terminology came about. $\endgroup$
    – pjs36
    May 14, 2015 at 14:26
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    $\begingroup$ So I guess we pick $r$ and draw the edge you have labelled as $r$. Then the hypotenuse of the resulting triangle is $r^2$? Can you please post a proof of that? Does this depend on the length of the other two edges of the first right triangle (the leftmost one with one leg labelled $1$). You haven't labelled them and they aren't uniquely determined by one leg being length one. Thank you. $\endgroup$ May 14, 2015 at 14:37
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    $\begingroup$ @GregoryGrant The triangles are all similar, that uniquely determines everything. The second triangle from the bottom is drawn first and then the first triangle is drawn similar to the second and so on... $\endgroup$
    – Asvin
    May 14, 2015 at 14:47
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    $\begingroup$ Is there any useful information that can be read off this figure? What would it look like for the interesting case when $r < 1$? $\endgroup$
    – Rob Arthan
    May 14, 2015 at 19:16
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    $\begingroup$ @RobArthan If $r<1$, you draw the figure approaching the vertex rather than going away from it. $\endgroup$ May 15, 2015 at 9:13
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Let $ABC$ is a right triangle with right angle $\angle ABC$, then if we draw the height $BH$, we have $$ |BH|^2=|AH||CH| $$ The geometric mean comes from here...

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    $\begingroup$ Isn't the height of the right triangle just $|BC|$? $\endgroup$ May 14, 2015 at 13:10
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    $\begingroup$ Observe that the right angle is $\angle B$ $\endgroup$
    – k1.M
    May 14, 2015 at 13:11
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    $\begingroup$ Ah, so you mean drop a perpendicular from the point $B$ to its opposite side, then $H$ is the point of intersection? $\endgroup$ May 14, 2015 at 13:13
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    $\begingroup$ Yes...Exactly... $\endgroup$
    – k1.M
    May 14, 2015 at 13:14
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The arithmetic and geometric adjectives come from the Pythagoreans before the Christian Era. Apparently, the expression “geometric progression” comes from the “geometric mean” (Euclidean notion) of segments of length $a$ and $b$: it is the length of the side $c$ of a square whose area is equal to the area of the rectangle of sides $a$ and $b$. The construction of the geometric mean with ruler and compass is well known for high school students; it involves "multiplication" and not "addition".

enter image description here

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    $\begingroup$ This picture perfectly explains why the geometric mean is smaller then the arithmetic mean if $a\neq b$. and equal if a=b. Beautiful. By Thales' theorem its the same as the comment of the right triangle $\endgroup$
    – Libertas
    Aug 8, 2020 at 6:53
  • $\begingroup$ Thank you for your comment, dear friend. $\endgroup$
    – Piquito
    Aug 9, 2020 at 11:08
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My guess would be that geometric sequences arose as generalization of sequence $a,a^2,a^3,...$. Why is this geometric? Well, $a$ is the length (= 1-dimensional "volume") of line segment (= 1-dimensional hypercube) of "side" length $a$, $a^2$ is the area (= 2-dimensional "volume") of a square (= 2-dimensional hypercube) of side length $a$, $a^3$ is the volume (= 3-dimensional "volume") of a cube (= 3-dimensional hypercube) of side length $a$ and so on.

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Because geometric progressions are based on multiplication, and the most important geometric notion, namely, volume, arises from multiplication (length times width times height). The term “multiplicative” is not used because it already has a special meaning in Number Theory.

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