ALGEBRA II H/G @ ARITHMETIC & GEOMETRIC MEANS.

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Presentation transcript:

ALGEBRA II H/G @ ARITHMETIC & GEOMETRIC 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. SOLUTION : 37, _______, _______, _______, _______, 52 tn = t1 + (n – 1)d 37 + 3 = 40 = 37 + (6 – 1)d 40 + 3 = 43 52 = 37 + 5d 43 + 3 = 46 15 = 5d 46 + 3 = 49 3 = d

2) Insert 2 geometric means between 52 and 73. SOLUTION : 52, _______, _______, 73 tn = t1 • rn-1 52 • 1.12 = 58.24 = 52 • r4-1 58.24 • 1.12 = 65.23 73 = 52 • r3 1.4038 = r3 1.12 = r

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.

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 Just for fun, try this : The mean proportional between a and b is 75 • 1.5 = 112.5