SOLUTION: If the 3rd term of a gp is four times its 5th term, find the possible values of the common ratio

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Question 1208816: If the 3rd term of a gp is four times its 5th term, find the possible values of the common ratio
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.

If the 3rd term of a gp is four times its 5th term, we can write this equation

    a%5B5%5D = r%5E2%2Aa%5B3%5D,    (valid for any GP)

    a%5B3%5D = 4%2Aa%5B5%5D.     (because the 3rd term of the GP is four times its 5th term).


It implies

    a%5B3%5D = 4%2A%28r%5E2%2Aa%5B3%5D%29%29 = %284r%5E2%29%2Aa%5B3%5D.


If  a%5B3%5D =/= 0,  then from previous equation we have

    4*r^2 = 1,  r^2 = 1%2F4,


which has two solutions  r = sqrt%281%2F4%29 = +/- 1%2F2.


ANSWER.   In regular case, when the GP is not zero-sequence,  the common ratio is either  1%2F2  or  -1%2F2.

          If case of zero-sequence, the common ratio can be any real number.

Solved.


Thanks to @greenestamps for noticing my error in previous version.

Now you see the corrected solution.



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


For tutor @ikleyn....

You will want to revise your response -- the problem says that the 3rd term is 4 times the 5th term, not the other way around.