Geometric means are the terms between non-consecutive terms of a geometric sequence. If A1, A2, .....An-1, An is a geometric sequence, then the numbers A2,.....,An-1 are the geometric means between A1 and An. Three examples are provided to illustrate how to find geometric means between given numbers: inserting geometric means between 1 and 81, between 8 and 512, and finding the single geometric mean between 5 and 8.

1. Geometric Mean
Ms. G. Martin

2. The terms between any two non-consecutive terms of a
geometric sequence are called geometric means.
If A1, A2, …..An-1, An is a geometric sequence, then the numbers
A2,…..,An-1 are the geometric means between A1 and An.

3. Example 1. Insert three geometric means between 1 and
81
Steps Solution
1. Identify the given information
(1, ___, ___, ___, 81)
Here, A1 = 1 and An (which is A5) =
81
2. Find the common ratio using
the formula An = A1rn-1
.
A1 = 1 and A5 = 81, we have
An = A1rn – 1
81 = (1)r4
81 = r4
4
81 = r
3 = r

4. 3. Solve for the three geometric
means using the common ratio.
1, ___, ___, ___, 81
A1 = 1
A2 = 1(3) = 3
A3 = 3(3) = 9
A4 = 9(3) = 27
So, the three geometric means between 1 and 81 are 3, 9, and 27.

5. Example 2. Insert five geometric means between 8 and 512.
Steps Solution
1. Identify the given information. A1 = 8 and A7 = 512
2. Solve the common ratio using the
formula
An = A1rn-1.
A1 = 8 and A1= 512, we have
An = A1rn-1
512 = 8(r)7 – 1
512 = 8r6
64 = r6
6
64 = r
2 = r

6. 3. Solve for the five geometric means
using the common ratio.
8, A2, A3, A4, A5, A6, 512
A1 = 8
A2 = 8(2) = 16
A3 = 16(2) = 32
A4 = 32(2) = 64
A5 = 64(2) = 128
A6 = 128(2)= 256
A7 = 512
So, the five geometric means between 8 and 512 are 16, 32, 64, 128, 256.

7. Example 3. Find the geometric mean between 5 and 8.
Steps Solution
1. Identify the given information Hence m = 5 and n = 8
2. Solve the geometric mean using
the formula of 𝑚𝑛 .
𝑚𝑛 = 5. 8 = 40
= 4. 10
= 2 10
So the geometric mean between 5 and 8 is 2 10.