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Is there a reason why a series of the form $1+2+4+8+16+...$ called a Geometric series and another of the form $1+3+5+7+9+...$ called an arithmetic series?

Martin Sleziak
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Solidification
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    This just shifts the problem, but $a_n$ is either the arithmetic or the geometric mean of $a_{n-1}$ and $a_{n+1}$. – Hagen von Eitzen Jan 14 '20 at 13:36
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    What is geometric about it? I don't see any connection to geometry. – Solidification Jan 14 '20 at 13:37
  • That the height to the hypotenuse of a right angled triangle is the geometric mean of the segments in which its base divides the hypotenuse. And you can also ask about the harmonic mean $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ and harmonic progression $1,\frac{1}{2},\frac{1}{3},...$ – MoonLightSyzygy Jan 14 '20 at 13:47
  • For what it's worth, the geometric mean of two numbers $a$ and $b$ is the side length of the square with the same area as the rectangle with sides $a$ and $b$, i.e. it comes up when "squaring a rectangle". – Klaus Jan 14 '20 at 13:48
  • I think @HagenvonEitzen has a very relevant justification since the geometric mean of two number can be constructed easily using a ruler and a compass. ($c$ is the geometric mean of $a$ and $b$ on this image). – nicomezi Jan 14 '20 at 13:54
  • In arthmetic we are interested in quantity. Very typically this involves adding things together. *very* informally and colloquially. In geometry we are interested in area. Very typically this involves multiplying things (sides) together. – fleablood Jan 14 '20 at 16:19
  • In an arithmetic sequence: $a_n = \frac {a_{n-1} + a_{n+1}}2$ so each term is the arithmetic mean between the other terms around it. In a geometric sequence $s_n = \sqrt{s_{n-1}s_{n+1}}$. This is called the "geometric mean". Why? Because it describes how you average the sides of a rectangle while maintaining the area. If you have a rectangle with two sides, $a$ and $b$, how do you "average" the sides to get a square with the same area? Answer: Area=$ab$, so $s^2=ab$ and $s =\sqrt{ab}$. And that's a GEOMETRIC problem. – fleablood Jan 14 '20 at 16:25

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Here is an arithmetic problem: If one person has $a$ items and another person has $b$ items, how to you redistribute them so that each person has the same amount?

Solution: If $n$ is the amount we want then we need Total: $2n = a+b$ and so $n=\frac{a+b}2$. That is called the arithmetic mean.

Here is a geometric problem: If a rectangle has sides $a$ and $b$, how to you readjust the sides to make a square with the same area.

Solution: If $s$ is the side of the square then we need Area: $s^2 =ab$ so $s = \sqrt{ab}$. This is called the geometric mean.

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Let's define sequence $a_0, a_1, a_2,.....$ where $a_n$ is the arithmetic mean between $a_{n-1}$ and $a_{n+1}$. Then we will notice that there is a $d = a_n - a_{n-1} = a_{n+1} - a_n$ and we can define the sequence as $a_0 = a$ and $a_{i+1} =a_i + d$. We call this an arithmetic sequence.

Let's define sequence $b_0, b_1, b_2,.....$ where $b_n$ is the geometric mean between $a_{n-1}$ and $a_{n+1}$. The we will notice that there is an $r=\frac{a_n}{a_{n-1}}=\frac {a_{n+1}}{a_n}$ and we can define the sequence as $b_0 = b$ and $b_{i+1}=b_i*r$. We call this a geometric sequence.

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A series is a sum of the terms of a sequence so the sum of an arithmetic/geometric sequence is called an arithmetic/geometric series.

fleablood
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