Question

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

Answer 1

Geometric Sequence

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

`{: (1st" term", =a), (2nd" term", =ar), (3rd " term", =ar^2 ), (vdots, ), (nth " term", =ar^(n-1) ) :}`

Where:

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

We typically denote the `nth` term by `u_n` and write the sequence in the form:

`{ u_n }` Or `{ a, ar, ar^2, ar^3, ... }`

Example 1:

`a=1, r=2 => u_n=2^n` Generating `{1,2,4,8,16, ...}`

Geometric Series

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

`S_2 = a+ar`
`S_3 = a+ar +ar^2`
`vdots`
`S_n = a+ar +ar^2 + ... + ar^(n-1)`

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

`S_3 = sum_(i=0)^2 ar^i`

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

`S_n \ \ = a+ar+ar^2 + ar^3 + ... + ar^(n-2) \ + ar^(n-1)` ..... [A}
`rS_n = \ \ \ \ \ \ \ ar+ar^2+ar^3 + ... + a2r^(n-1) + ar^(n) \ \ \ \ \` ..... [B}

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

`S_n - rS^n = a + 0 + 0 + ... + 0 - ar^n`
`:. S_n(1-r)n = a - ar^n`
`:. S_n = a(1 - r^n) / (1-r)`

Example 2

`a=1, r=2 => S_4=1+2+4+8=15`

Or with the formula:

`S_4 = (1)(1 - 2^4) / (1-2) = -15/-1 = 15`

Geometric Mean

is the `nth` root of the product of `n` numbers. If we have numbers:

`x_1, x_2, ... x_n`

Then:

`GM = root(n)((x_1 * x_2 * ...* x_n))`

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

`GM = root(n)(prod_(i=1)^n x_i )`

Example 3

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

`GM = sqrt(1*2) = sqrt(2)`

Example 4

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

`GM = root(3)(1*2*4) = root(3)(8) = 2`