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

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#