Answer
for $r=\sqrt 6$:
$a_1=2$
$a_2=2\sqrt 6$
$a_3=12$
$a_4=12\sqrt 6$
$a_5=72$
for $r=-\sqrt 6$:
$a_1=2$
$a_2=-2\sqrt 6$
$a_3=12$
$a_4=-12\sqrt 6$
$a_5=72$
Work Step by Step
$a_1=2,~~a_3=12$
The nth term of a geometric sequence:
$a_n=a_1r^{n-1}$
$a_3=a_1r^{3-1}$
$12=2r^2$
$6=r^2$
$r=\sqrt 6$ or $r=-\sqrt 6$
for $r=\sqrt 6$:
$a_2=a_1r^{2-1}=2\sqrt 6$
$a_3=12$
$a_4=a_1r^{4-1}=2(\sqrt 6)^3=2(6\sqrt 6)=12\sqrt 6$
$a_5=a_1r^{5-1}=2(\sqrt 6)^4=2(36)=72$
for $r=-\sqrt 6$:
$a_2=a_1r^{2-1}=2(-\sqrt 6)=-2\sqrt 6$
$a_3=12$
$a_4=a_1r^{4-1}=2(-\sqrt 6)^3=2(-6\sqrt 6)=-12\sqrt 6$
$a_5=a_1r^{5-1}=2(-\sqrt 6)^4=2(36)=72$
