Geometric Sequence: Mean

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
is the answer.

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
is the answer.