a)
[tex]q=\frac{a_3}{a_2}=\frac{\sqrt{3}}{3}\\[/tex]
b)
[tex]a_6=a^2*q^4\\\\a_6=3*(\frac{\sqrt{3}}{3})^4\\\\a_6=3*\frac{9}{81}=\frac{1}{3}\\[/tex]
c)
[tex]S_8=3\sqrt{3}*\frac{1-\frac{\sqrt{3}}{3}^8}{1-\frac{\sqrt{3}}{3}}\\\\S_8=3\sqrt{3}*\frac{1-\frac{1}{81}}{1-\frac{\sqrt{3}}{3}}\\\\S_8=3\sqrt{3}*\frac{\frac{80}{81}}{1-\frac{\sqrt{3}}{3}}\\\\S_8=\frac{3\sqrt{3}*\frac{80}{81}}{1-\frac{\sqrt{3}}{3}}}\\\\S_8=\frac{\frac{80}{27}\sqrt{3}}{\frac{3-\sqrt{3}}{3}}\\\\S_8=\frac{80\sqrt{3}}{9(3-\sqrt{3})}\\\\S_8=\frac{80\sqrt{3}}{9(3-\sqrt{3})}*\frac{27+9\sqrt{3}}{27+9\sqrt{3}}\\\\S_8=\frac{241\sqrt{3}}{54}\\[/tex]
d)
[tex]a_n=a_1*q^{n-1}\\\\a_n=3\sqrt{3}*(\frac{\sqrt{3}}{3})^{n-1}\\\\a_n=3^{1+\frac{1}{2}}*(3^{\frac{1}{2}-1})^{n-1}\\\\a_n=3^{\frac{3}{2}}*(3^{-\frac{1}{2}})^{n-1}\\\\a_n=3^{\frac{3}{2}}*3^{-\frac{1}{2}n+\frac{1}{2}}\\\\a_n=3^{\frac{3}{2}-\frac{1}{2}n+\frac{1}{2}}\\\\a_n=3^{\frac{4}{2}-\frac{1}{2}n}\\[/tex]
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a)
[tex]q=\frac{a_3}{a_2}=\frac{\sqrt{3}}{3}\\[/tex]
b)
[tex]a_6=a^2*q^4\\\\a_6=3*(\frac{\sqrt{3}}{3})^4\\\\a_6=3*\frac{9}{81}=\frac{1}{3}\\[/tex]
c)
[tex]S_8=3\sqrt{3}*\frac{1-\frac{\sqrt{3}}{3}^8}{1-\frac{\sqrt{3}}{3}}\\\\S_8=3\sqrt{3}*\frac{1-\frac{1}{81}}{1-\frac{\sqrt{3}}{3}}\\\\S_8=3\sqrt{3}*\frac{\frac{80}{81}}{1-\frac{\sqrt{3}}{3}}\\\\S_8=\frac{3\sqrt{3}*\frac{80}{81}}{1-\frac{\sqrt{3}}{3}}}\\\\S_8=\frac{\frac{80}{27}\sqrt{3}}{\frac{3-\sqrt{3}}{3}}\\\\S_8=\frac{80\sqrt{3}}{9(3-\sqrt{3})}\\\\S_8=\frac{80\sqrt{3}}{9(3-\sqrt{3})}*\frac{27+9\sqrt{3}}{27+9\sqrt{3}}\\\\S_8=\frac{241\sqrt{3}}{54}\\[/tex]
d)
[tex]a_n=a_1*q^{n-1}\\\\a_n=3\sqrt{3}*(\frac{\sqrt{3}}{3})^{n-1}\\\\a_n=3^{1+\frac{1}{2}}*(3^{\frac{1}{2}-1})^{n-1}\\\\a_n=3^{\frac{3}{2}}*(3^{-\frac{1}{2}})^{n-1}\\\\a_n=3^{\frac{3}{2}}*3^{-\frac{1}{2}n+\frac{1}{2}}\\\\a_n=3^{\frac{3}{2}-\frac{1}{2}n+\frac{1}{2}}\\\\a_n=3^{\frac{4}{2}-\frac{1}{2}n}\\[/tex]