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 an = a1 + (n – 1) d 8 = 29 + 3d -21 = 3d -7 = d 29, 22, 15, 8

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 = 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 an = a1 + (n – 1) d 17 = 5 + 2d 12 = 2d 6 = d 5, 11, 17

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 7

Find the real number solution. 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 -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.