Download Arithmetic and Geometric Means Arithmetic and Geometric Means

Arithmetic and Geometric
Means
OBJ: •
Find arithmetic and geometric
means
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, . . .
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
a = a + (n – 1) d
8 = 29 + 3d
-21 = 3d
-7 = d
29, 22, 15, 8
n
1
As shown in the example below, you can
find any specified number of arithmetic
means between two given numbers.
EX:  Find the five 30,__,__,__,__,__, 21
arithmetic means a = a + (n – 1) d
between 30 and 21.
n
1
21 = 30 + 6d
-9 = 6d
-1.5 = d
30, 28.5, 27, 25.5, 24,
22.5,21
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
a = a + (n – 1) d
17 = 5 + 2d
12 = 2d
6=d
5, 11, 17
n
1
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.
-8 + 22
2
14
2
7
Find the real number solution.
1. r2 = 5
r = ±5
2. r3 = -8
r = -2
3. r3 = _ 64
125
r = -4
5
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 96 . . .
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 24.
8
-3, ____, ____, 24
8
l = a •rn – 1
24 = -3 •r 3
8
-64 = r 3
-4 = r
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
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
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.