# Geometric Means

How to find the geometric means: examples and their solutions.

## Example 1

Geometric means are the terms

that form an geometric sequence

with the boundary terms.

Write 6, four blanks, and 192.

The values of the blanks are

6*r*, 6*r*^{2}, 6*r*^{3}, and 6*r*^{4}.

So 192 = 6*r*^{5}.

Geometric sequences

6*r*^{5} = 192

So *r* = 2.

Find the values of the blanks.

(1st blank) = 6⋅2 = 12

(2nd blank) = 12⋅2 = 24

(3rd blank) = 24⋅2 = 48

(4rd blank) = 48⋅2 = 96

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

## Example 2

Write 5, three blanks, and 405.

The values of the blanks are

5*r*, 5*r*^{2}, and 5*r*^{3}.

So 405 = 5*r*^{4}.

Geometric sequences

5*r*^{4} = 405

So *r* = ±3.

Find the values of the blanks.

(1st blank) = 5⋅(±3) = ±15

(2nd blank) = (±15)⋅(±3) = +45

(3rd blank) = 45⋅(±3) = ±135

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