ALGEBRA II H/GALGEBRA II H/G@@

ARITHMETIC &ARITHMETIC &GEOMETRIC MEANSGEOMETRIC MEANS

MEAN : average

ARITHMETIC or GEOMETRIC MEANS :

between two numbers are the terms which form an arithmetic sequence or a geometric sequence between the two given terms.

1) Insert 4 arithmetic means between 37 and 52.

tn = t1 + (n – 1)d 37 + 3 = 40

52= 37 + (6 – 1)d 40 + 3 = 43

52 = 37 + 5d 43 + 3 = 46

15 = 5d 46 + 3 = 49

3 = d

SOLUTION :

37, _______, _______, _______, _______, 52

2) Insert 2 geometric means between 52 and 73.

tn = t1 • rn-1 52 • 1.12 = 58.24

73= 52 • r4-1 58.24 • 1.12 = 65.23

73 = 52 • r3

1.4038 = r3

1.12 = r

SOLUTION :

52, _______, _______, 73

3) Insert 3 geometric means between 6 and 96 if complex numbers are allowed.

SOLUTION : 6, _______, _______, _______, 96

tn = t1 • rn-1

96 = 6 • r5-1

16 = r4

Now, take the square root of both sides.

4

2

16= r±4 = r

2r = 4r =±2

2r = -4r =±2i

If r = 2 : 6 • 2 = 12, 12 • 2 = 24

24 • 2 = 48

If r = -2 : 6 • -2 = -12,

-12 • -2 = 24

24 • -2 = -48

If r = 2i : 6 • 2i = 12i,

12i • 2i = -24

-24 • 2i = -48i

If r = -2i : 6 • -2i = -12i, -12i •- 2i = 2424 • -2i = -48i

4) Find the mean proportional of 75 and 168.75.MEAN PROPORTIONAL : means insert one geometric mean between the two terms. The geometric mean, by definition, is positive.SOLUTION : 75, _______, 168.75

tn = t1 • rn-1

168.75 = 75 • r3-1

2.25 = r2 22.25 = r1.5 = r

75 • 1.5 = 112.5

Just for fun, try this :

The mean proportional between a and b isab

75 • 168.75 =12656.25 =112.5