SOLUTION: Insert three geometric means between 1/9 and 27

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Question 1106813: Insert three geometric means between 1/9 and 27
Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


The two given numbers are 1%2F9+=+3%5E%28-2%29 and 27+=+3%5E3.

Are you sure you have shown the problem correctly? Inserting FOUR geometric means would be easy; inserting three is ugly.

To insert four geometric means, we would need to have
27+=+%281%2F9%29%2Ar%5E5
3%5E3+=+%283%5E%28-2%29%29%2Ar%5E5
3%5E5+=+r%5E5
3+=+r

The sequence including the four geometric means would then be
1/9, 1/3, 1, 3, 9, 27.

To insert three geometric means, we need to have
27+=+%281%2F9%29%2Ar%5E4
3%5E3+=+%283%5E%28-2%29%29%2Ar%5E4
3%5E5+=+r%5E4
r+=+3%5E%285%2F4%29
The sequence including only three geometric means would then be
3^(-2), 3^(-3/4), 3^(1/2), 3^(7/4), 3^3.