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2016-07-27T17:52:25+08:00

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Problem E:  Three positive numbers form a sequence.  If the geometric mean of the first two numbers is 6, and the geometric means of the last two numbers is 24, find the three numbers and their common ratio,

The sequence would be:       a₁,  6,  a₂,  24,  a₃

Step 1:  Find a₂

a
₂ = +/-  \sqrt{(6)(24)}
a₂ = 12

Since the three missing terms are positive, we choose a₂ = +12.

Step 2:  Divide a₂ by the geometric mean (6)  between a₁ and a₂:
r = 12/6
r = 2

Step 3:  Find the first (a₁) and the last (a₃) numbers.

a₁ = 6/2
a₁ = 3

a₃ = 24 (2)
a₃ = 48

Step 4:  Find the common ratio of the geometric sequence 3, 12, 48:

12/3 = 4
48/12 = 4
r = 4

ANSWER:
The geometric sequence is 3, 12, 48
The common ratio is 4.

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