# Geometric Sequence: Mean

How to find the means of a geometric sequence: definition, 2 examples, and their solutions.

## Definition

### Definition

The geometric means are the terms
that form a geometric sequence
with the first and the last term.

In this figure,
a1, the blanks, and a4
form a geometric sequence.
Then the blanks are the geometric means.

## Example 1

### Solution

It says
find the four geometric means
between 6 and 192.

So write 6, four blanks (the geometric means), and 192.

Write the common ratio ×r
between the terms.

192 is found by
multiplying the first term 6
and r 5 times.

So 192 = 6⋅r5.

Write 6⋅r5 = 192.

Divide both sides by 6.

Then r5 = 32.

32 = 25

r5 = 25

Fifth root both sides.

Then r = 2.

nth Root

a = 6
r = 2

Then find the four geometric means
by multiplying r = 2.

6⋅2 = 12

This 12 is the first geometric mean.

12⋅2 = 24

This 24 is the second geometric mean.

24⋅2 = 48

This 48 is the third geometric mean.

48⋅2 = 96

This 96 is the fourth geometric mean.

So the four geometric means are
12, 24, 48, 96.

So
12, 24, 48, 96

## Example 2

### Solution

It says
find the three geometric means
between 5 and 405.

So write 5, three blanks (the geometric means), and 405.

Write the common ratio ×r
between the terms.

405 is found by
multiplying the first term 5
and r 4 times.

So 405 = 5⋅r4.

Write 5⋅r4 = 405.

Divide both sides by 5.

Then r4 = 81.

81 = 34

r4 = 34

Fourth root both sides.

Then r = ±3.

Fourth root is an even root.
So write ±.

nth Root

See the two cases.

Case 1: r = 3

a = 5
r = 3

Then find the three geometric means
by multiplying r = 3.

5⋅3 = 15

This 15 is the first geometric mean.

15⋅3 = 45

This 45 is the second geometric mean.

45⋅3 = 135

This 135 is the third geometric mean.

So, when r = 3,
the three geometric means are
15, 45, 135.

See the next case.

Case 2: r = -3

a = 5
r = 3

Then find the three geometric means
by multiplying r = -3.

5⋅-3 = -15

This -15 is the first geometric mean.

-15⋅(-3) = +45

This 45 is the second geometric mean.

45⋅(-3) = -135

This -135 is the third geometric mean.

So, when r = -3,
the three geometric means are
-15, 45, -135.

Case 1: r = 3
The three geometric means are
15, 45, 135.

Case 2: r = -3
The three geometric means are
-15, 45, -135.

So the three geometric means are
±15, 45, ±135.

So
±15, 45, ±135