1. Arithmetic and Geometric Means
OBJ: •
Find arithmetic and geometric means

2. Arithmetic means are the terms between two given terms of an arithmetic progression or sequence.
For example, three arithmetic means between 2 and 18 in the progression below are 6, 10, and 14 since 2, 6, 10, 14, 18, is an arithmetic progression.
2, 6, 10, 14, 18, . . .

3. As shown in the example below, you can find any specified number of arithmetic means between two given numbers.
EX:  Find two arithmetic means between 29 and 8.
29, ____, ____, 8
an = a1 + (n – 1) d
8 = d
-21 = 3d
-7 = d
29, 22, 15, 8

4. As shown in the example below, you can find any specified number of arithmetic means between two given numbers.
EX:  Find the five arithmetic means
between 30 and 21.
30,__,__,__,__,__, 21
an = a1 + (n – 1) d
21 = d
-9 = 6d
-1.5 = d
30, 28.5, 27, 25.5, 24, 22.5,21

5. As shown in the example below, you can find any specified number of arithmetic means between two given numbers.
EX:  Find the one arithmetic mean
between 5 and 17.
5, ____, 17
an = a1 + (n – 1) d
17 = 5 + 2d
12 = 2d
6 = d
5, 11, 17

6. Since this is the same
as the average of 5
and 17, it easier to use
the formula: x + y. 2
which is called the
arithmetic mean of the
real numbers x and y.
EX:  Find the arithmetic mean of
-8 and 22. 2 14 7

7. Find the real number solution.
125
r = -4 5

8. Geometric means are the terms between two given terms of a geometric progression or sequence.
For example, four geometric means between 3 and 96 in the progression below are 6, 12, 24, and 48 since 3, 6, 12, 24, 48, 96, is a geometric progression.
3, 6, 12, 24, 48, and

9. As shown in the example below, you can find any specified number of geometric means between two given numbers.
EX:  Find the two real geometric means between –3 and
-3, ____, ____, 24 8
l = a •rn – 1
24 = -3 •r 3
-64 = r 3
-4 = r

10. As shown in the example below, you can find any specified number of geometric means between two given numbers.
EX:  Find three geometric means
between 32 and 2.
32, ____, ____, ____, 2
l = a •rn – 1
2 = 32 •r4
1 = r4 16
± 1 2

11. As shown in the example below, you can find any specified number of geometric means between two given numbers.
EX:  Find one geometric mean between 5 and 10
5, ____, 10
l = a •rn – 1
10 = 5 •r2
2 = r2
±2

12. The geometric mean (mean proportional) of the real numbers x and y (xy > 0) is
 xy or –  xy .
EX:  Find the positive geometric mean of
4 and 8.