Any reason why the geometric and the arithmetic series are so named?

Question

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?

Answer 1

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.