1. ALGEBRA II H/G @
ARITHMETIC &
GEOMETRIC MEANS

2. 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.

3. 1) Insert 4 arithmetic means between 37 and 52.
SOLUTION :
37, _______, _______, _______, _______, 52
tn = t1 + (n – 1)d = 40
= 37 + (6 – 1)d = 43
52 = d = 46
15 = 5d = 49
3 = d

4. 2) Insert 2 geometric means between 52 and 73.
SOLUTION :
52, _______, _______, 73
tn = t1 • rn • 1.12 = 58.24
= 52 • r • 1.12 = 65.23
73 = 52 • r3
= r3
1.12 = r

5. SOLUTION : 6, _______, _______, _______, 96 tn = t1 • rn-1
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
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 = 24
24 • -2i = -48i
Now, take the square root of both sides.

6. 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, _______,
tn = t1 • rn-1
= 75 • r3-1
= r2
Just for fun, try this :
The mean proportional between a and b is
75 • 1.5 = 112.5