SOLUTION: The first three terms of a geometric sequence are lnx^16, lnx^8, lnx^4, for x>0. Find the common ratio.

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Question 1192803: The first three terms of a geometric sequence are lnx^16, lnx^8, lnx^4, for x>0. Find the common ratio.
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
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The first three terms of a geometric sequence are lnx^16, lnx^8, lnx^4, for x>0. Find the common ratio.
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The 1-st term is  ln(x^16) = 4*ln(x^4).


The 2-nd term is  ln(x^8) = 2*ln(x^4).


The 3-rd term is  ln(x^4).


The ratio  a%5B2%5D%2Fa%5B1%5D = 1%2F2.


The ratio  a%5B3%5D%2Fa%5B2%5D = 1%2F2.


So the progression is geometric and its common ratio is  1%2F2.    ASNSWER

Solved and explained.

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The side note :   in this problem,  x > 0   IS  NOT   a necessary condition.  The necessary condition is   | x | =/= +/-1.



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The common ratio is the second term divided by the first term:

ln%28x%5E8%29%2Fln%28x%5E16%29
ln%28x%5E8%29%2Fln%28%28x%5E8%29%5E2%29
ln%28x%5E8%29%2F%282%2Aln%28x%5E8%29%29=1%2F2

The common ratio is 1/2.

By the way.... The comment from the other tutor that the "for x>0" is not needed in the problem is incorrect. The sequence is

ln(x^16), ln(x^8), ln(x^4), ln(x^2), ln(x), ln(x^(1/2)), ln(x^(1/4)),...

x>0 IS required for this sequence, starting with the 5th term, ln(x).