Answer
$3^{\frac{1}{2}},\ 3^{\frac{3}{4}},\ 3^{\frac{7}{8}},\ 3^{\frac{15}{16}},\ 3^{\frac{31}{32}},\ 3^{\frac{63}{64}},\ 3^{\frac{127}{128}}$
Work Step by Step
$a_{1}=\sqrt{3}=3^{1/2}\qquad a_{n}=\sqrt{3a_{n-1}}$
$a_{2}=\sqrt{3a_{1}}=\sqrt{3\cdot 3^{1/2}}=\sqrt{3^{3/2}}=3^{3/4}$,
$a_{3}=\sqrt{3a_{2}}=\sqrt{3\cdot 3^{3/4}}=\sqrt{3^{7/4}}=3^{7/8}$,
$a_{4}=\sqrt{3a_{3}}=\sqrt{3\cdot 3^{7/8}}=\sqrt{3^{15/8}}=3^{15/16}$,
$a_{5}=\sqrt{3a_{4}}=\sqrt{3\cdot 3^{15/16}}=\sqrt{3^{31/16}}=3^{31/32}$,
$a_{6}=\sqrt{3a_{5}}=\sqrt{3\cdot 3^{31/32}}=\sqrt{3^{63/32}}=3^{63/64}$,
$a_{7}=\sqrt{3a_{6}}=\sqrt{3\cdot 3^{63/64}}=\sqrt{3^{127/64}}=3^{127/128}$.
