Geometric Means

Geometric Means

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

Example 1

Find the four geometric means between 6 and 192.

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
6r, 6r2, 6r3, and 6r4.

So 192 = 6r5.

Geometric sequences

6r5 = 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

Find the three geometric means between 5 and 405.

Write 5, three blanks, and 405.

The values of the blanks are
5r, 5r2, and 5r3.

So 405 = 5r4.

Geometric sequences

5r4 = 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.