SOLUTION: three numbers form a geometric progression if we double the middle number we get an arithematic progression the common ratio of the geomatric progression is ?

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Question 1037823: three numbers form a geometric progression if we double the middle number we get an arithematic progression the common ratio of the geomatric progression is ?
Found 2 solutions by stanbon, ikleyn:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
three numbers form a geometric progression if we double the middle number we get an arithmetic progression the common ratio of the geometric progression is ?
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Original Progression:: a, ar, ar^2
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New Progression:: a, 2ar, ar^2
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Common ratio:: Divide 2nd term by 1st to get:: (2ar)/a = 2r
Divide the 3rd term by the 2nd to get:: (ar^2)/(2ar) = r/2
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If 2r = r/2 , 4r = r
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Comment:: Your problem statement doesn't make much sense.
Cheers,
Stan H.
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Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.
three numbers form a geometric progression if we double the middle number we get an arithmetic progression.
the common ratio of the geometric progression is ?
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The original geometric progression:  a,  ar,  ar%5E2.

The new sequence is:  a,  2ar,  ar%5E2.

The new progression is arithmetic.  It means that  a%5B3%5D - a%5B2%5D = a%5B2%5D - a%5B1%5D.

In other words,  ar%5E2+-+2ar = 2ar+-+a.

Then  a%5E2+-+4ar+%2B+a = 0, --->
r%5E2+-+4r+%2B+1 = 0,
r%5B1%5D = 1+%2B+sqrt%283%29   (positive),
r%5B2%5D = 1+-+sqrt%283%29   (negative).

The previous solution was wrong.