# What is the difference between a Geometric Sequence, Geometric Series and a Geometric Mean?

Aug 12, 2017

Geometric Sequence

is an ordered list of numbers that obey a relationship of the form:

$\left.\begin{matrix}1 s t \text{ term" & =a \\ 2nd" term" & =ar \\ 3rd " term" & =ar^2 \\ vdots & \null \\ nth " term} & = a {r}^{n - 1}\end{matrix}\right.$

Where:

$a =$ First Term
$r =$ factor between terms (or the "common ratio")

We typically denote the $n t h$ term by ${u}_{n}$ and write the sequence in the form:

$\left\{{u}_{n}\right\}$ Or $\left\{a , a r , a {r}^{2} , a {r}^{3} , \ldots\right\}$

Example 1:

$a = 1 , r = 2 \implies {u}_{n} = {2}^{n}$ Generating $\left\{1 , 2 , 4 , 8 , 16 , \ldots\right\}$

Geometric Series

is a sum of consecutive terms of a geometric sequence, so for example:

${S}_{2} = a + a r$
${S}_{3} = a + a r + a {r}^{2}$
$\vdots$
${S}_{n} = a + a r + a {r}^{2} + \ldots + a {r}^{n - 1}$

We condense the series using "sigma" notation, where the greek symbol $\sum$ denotes sum

${S}_{3} = {\sum}_{i = 0}^{2} a {r}^{i}$

And in fact we can derive a formula for the sum of the first $n$ terms:

${S}_{n} \setminus \setminus = a + a r + a {r}^{2} + a {r}^{3} + \ldots + a {r}^{n - 2} \setminus + a {r}^{n - 1}$ ..... [A}
$r {S}_{n} = \setminus \setminus \setminus \setminus \setminus \setminus \setminus a r + a {r}^{2} + a {r}^{3} + \ldots + a 2 {r}^{n - 1} + a {r}^{n} \setminus \setminus \setminus \setminus \setminus$ ..... [B}

If we take the difference [A] - [B], then almost all the terms cancel

${S}_{n} - r {S}^{n} = a + 0 + 0 + \ldots + 0 - a {r}^{n}$
$\therefore {S}_{n} \left(1 - r\right) n = a - a {r}^{n}$
$\therefore {S}_{n} = a \frac{1 - {r}^{n}}{1 - r}$

Example 2

$a = 1 , r = 2 \implies {S}_{4} = 1 + 2 + 4 + 8 = 15$

Or with the formula:

${S}_{4} = \left(1\right) \frac{1 - {2}^{4}}{1 - 2} = - \frac{15}{-} 1 = 15$

Geometric Mean

is the $n t h$ root of the product of $n$ numbers. If we have numbers:

${x}_{1} , {x}_{2} , \ldots {x}_{n}$

Then:

$G M = \sqrt[n]{\left({x}_{1} \cdot {x}_{2} \cdot \ldots \cdot {x}_{n}\right)}$

We condense the product notation using the "Pi" notation, where the greek symbol $\Pi$ denotes product

$G M = \sqrt[n]{{\prod}_{i = 1}^{n} {x}_{i}}$

Example 3

The geometric mean of $1$ and $2$ is:

$G M = \sqrt{1 \cdot 2} = \sqrt{2}$

Example 4

The geometric mean of $1 , 2 , 4$ is:

$G M = \sqrt{1 \cdot 2 \cdot 4} = \sqrt{8} = 2$