# Geometric Means

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

## Example 1: Even Means

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 6*r*.

The second blank is 6*r*^{2}.

The third blank is 6*r*^{3}.

The fourth blank is 6*r*^{4}.

And 192 is 6*r*^{5}.

Geometric sequences

192 is 6*r*^{5}.

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

Divide both sides by 6.

Then *r*^{5} = 32.

32 = 2^{5}

So *r* = 5.

The exponent of *r*^{5} 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

Write 5, three blanks, and 405.

These five terms are a geometric sequence.

Set the common ratio as *r*.

Then the first blank is 5*r*.

The second blank is 5*r*^{2}.

The third blank is 5*r*^{3}.

And 405 is 5*r*^{4}.

Geometric sequences

405 is 5*r*^{4}.

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

Divide both sides by 5.

Then *r*^{4} = 81.

81 = 3^{4}

So *r* = 3.

The exponent of *r*^{4} 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.