The sequences of the given terms are required.
The sequences are [tex]27,18,12,8,\dfrac{16}{3}\ldots[/tex]
and [tex]-27,18,-12,8,-\dfrac{16}{3}\ldots[/tex]
The terms are
[tex]a_2=18\\\Rightarrow ar^{2-1}=18\\\Rightarrow ar=18[/tex]
[tex]a_4=8\\\Rightarrow ar^{4-1}=8\\\Rightarrow ar^3=8[/tex]
Dividing the equations
[tex]\dfrac{a_2}{a_4}=\dfrac{18}{8}\\\Rightarrow \dfrac{ar}{ar^3}=\dfrac{9}{4}\\\Rightarrow \dfrac{1}{r^2}=\dfrac{9}{4}\\\Rightarrow r=\pm\dfrac{2}{3}[/tex]
Common ratio
[tex]r=\dfrac{2}{3}[/tex]
[tex]a\times \dfrac{2}{3}=18\\\Rightarrow a=18\times \dfrac{3}{2}\\\Rightarrow a=27[/tex]
[tex]a_3=ar^2\\ =27(\dfrac{2}{3})^2\\ =12[/tex]
[tex]a_5=27(\dfrac{2}{3})^4\\ =\dfrac{16}{3}[/tex]
[tex]r=-\dfrac{2}{3}[/tex]
[tex]a\times -\dfrac{2}{3}=18\\\Rightarrow a=-18\times \dfrac{3}{2}\\\Rightarrow a=-27[/tex]
[tex]a_3=ar^2\\ =-27(-\dfrac{2}{3})^2\\ =-12[/tex]
[tex]a_5=-27(-\dfrac{2}{3})^4\\ =-\dfrac{16}{3}[/tex]
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