2

I am a bit confused on the meaning of "geometric". I believe I understand the concept of a "geometric mean" of a sequence vs. an "arithmetic" mean.

Now when I use a sequence to convert over to the partial sums of a series does the same concept apply here? I assume yes but not sure how. Do I add the partials and square root like the sequence ? Thank you

Jean-Claude Arbaut
  • 23,277
Sedumjoy
  • 1,569
  • 1
    Apparently they are discussed in Books VIII and IX of the Elements, see https://en.wikipedia.org/wiki/Geometric_progression#Relationship_to_geometry_and_Euclid's_work – Ian Jan 09 '18 at 18:05

1 Answers1

2

Because if $a>0$, $b>0$ and $c>0$ the geometric series then $b^2=ac$.

Now, for example, let $CD=b$, $AD=a$ and $BD=c$ in the $\Delta ABC$,

where $\measuredangle ACB=90^{\circ}$ and $CD$ is an altitude of the triangle.

There are very many another properties in geometry with $b^2=ac$.

Michael Rozenberg
  • 194,933
  • I don't understand how $a,b,c$ are defined in your first line. – Ian Jan 09 '18 at 18:16
  • 1
    $a$, $b$, $c$ is a geometric progression. Read about it. It's good. See here: https://en.wikipedia.org/wiki/Geometric_progression – Michael Rozenberg Jan 09 '18 at 18:18
  • I think it might be clearer to use more modern notation like $a(ar^2)=(ar)^2$. – Ian Jan 09 '18 at 18:21
  • $a = a$ and $b = ar$ and $c = ar^2$ so $ac = a^2r^2 = b^2$. – fleablood Jan 09 '18 at 18:21
  • So one way to think of it is $r$ is a proportion. $c_i$ is side of a square. Then $c_{i+1}$ will be the side of a rectangle that has the same area of the square if the other side of the rectangle is $r$ times the side of the square. – fleablood Jan 09 '18 at 18:24