Question 1208816
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<pre>

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

    {{{a[5]}}} = {{{r^2*a[3]}}},    (valid for any GP)

    {{{a[3]}}} = {{{4*a[5]}}}.     (because the 3rd term of the GP is four times its 5th term).


It implies

    {{{a[3]}}} = {{{4*(r^2*a[3]))}}} = {{{(4r^2)*a[3]}}}.


If  {{{a[3]}}} =/= 0,  then from previous equation we have

    4*r^2 = 1,  r^2 = {{{1/4}}},


which has two solutions  r = {{{sqrt(1/4)}}} = +/- {{{1/2}}}.


<U>ANSWER</U>.   In regular case, when the GP is not zero-sequence,  the common ratio is either  {{{1/2}}}  or  {{{-1/2}}}.

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

Solved.



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


Now you see the corrected solution.