Geometric Means

Geometric Means

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

Example 1: Even Means

Find the four geometric means between 6 and 192.

Geometric means are the terms
between the given terms
that form a geometric sequence.

To find the four geometric means between 6 and 192,
write 6, four blanks, and 192.

These six terms are a geometric sequence.

Set the common ratio as r.

Then the first blank is 6r.
The second blank is 6r2.
The third blank is 6r3.
The fourth blank is 6r4.
And 192 is 6r5.

Geometric sequences

192 is 6r5.

So 6r5 = 192.

Divide both sides by 6.

Then r5 = 32.

32 = 25

So r = 5.

The exponent of r5 is 5:
an odd exponent.
So you don't need to write [±]
in front of 2.

r = 2

To find the values of the blanks,
start from 6
and multiply 2.

The first blank is, 6⋅2, 12.

The second blank is, 12⋅2, 24.

The third blank is, 24⋅2, 48.

And the fourth blank is, 48⋅2, 96.

96⋅2 is 192,
which is the last term.
So you found the right terms.

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

Example 2: Odd Means

Find the three geometric means between 5 and 405.

Write 5, three blanks, and 405.

These five terms are a geometric sequence.

Set the common ratio as r.

Then the first blank is 5r.
The second blank is 5r2.
The third blank is 5r3.
And 405 is 5r4.

Geometric sequences

405 is 5r4.

So 5r4 = 405.

Divide both sides by 5.

Then r4 = 81.

81 = 34

So r = 3.

The exponent of r4 is 4:
an even exponent.
So you do need to write [±]
in front of 3.

Then you should think of two cases:
r = 3 and r = -3.

Case 1: r = 3

To find the values of the blanks,
start from 5
and multiply 3.

The first blank is, 5⋅3, 15.

The second blank is, 15⋅3, 15.

The third blank is, 45⋅3, 135.

And 135⋅3 is 405,
which is the last term.

Case 2: r = -3

Start from 5
and multiply -3.

The first blank is, 5⋅(-3), -15.

The second blank is, (-15)⋅(-3), +45.

The third blank is, 45⋅(-3), -135.

And (-135)⋅(-3) is +405,
which is the last term.

The blanks for case 1 are
15, 45, and 135.

And the blanks for case 2 are
-15, 45, and -135.

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

Keep in mind that
if the geometric means are even,
then some of the means have [±] signs.