Why is a geometric progression called so?

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

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

Here is a geometric figure illustrating the geometric progression $1,r,r^2,r^3,r^4,r^5,\ldots$:

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...