SOLUTION: Insert 3 geometric means between 256 and 1

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Question 1093781: Insert 3 geometric means between 256 and 1
Found 2 solutions by stanbon, MathTherapy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Insert 3 geometric means between 256 and 1
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Progression you want:: 256,256x,256x^2,256x^3,256x^4=1
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Solve for x, x^2, x^3
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256x^4 = 1
x^4 = 1/2^9
x = 1/2^(9/4)
x^2 = 1/2^(18/4)
x^3 = 1/2^(27/4)
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From that you can figure out the 3 geometric means.
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Cheers,
Stan H.
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Answer by MathTherapy(10549) About Me  (Show Source):
You can put this solution on YOUR website!

Insert 3 geometric means between 256 and 1
With r being the common ratio, we get the following 5-term sequence:  

Now, since the 5th term is 1, we get: 256r%5E4+=+1

r%5E4+=+1%2F256

r%5E4+=+1%5E4%2F4%5E4 

r%5E4+=+%281%2F4%29%5E4

r+=+1%2F4 ------ Exponents are equal and so are the bases

You can use this common ratio to get the 2nd, 3rd, and 4th terms of the series, which is the same as the 3 geometric means between 256 and 1.