Answer
The first five terms are:
$a_1= \sqrt2$
$a_2 = \dfrac{\sqrt{2\sqrt2}}{2}$
$a_3 = \dfrac{\sqrt[4]{2\sqrt2}}{2}$
$a_4 = \dfrac{\sqrt[8]{2\sqrt2}}{2}$
$a_5= \dfrac{\sqrt[16]{2\sqrt2}}{2}$
Work Step by Step
Use the given recursive formula to find the next four terms:
$a_1= \sqrt2$
$a_2 = \sqrt{\dfrac{a_1}{2}} = \sqrt{\frac{\sqrt2}{2}}=\sqrt{\frac{\sqrt2}{2}\cdot \frac{2}{2}}=\dfrac{\sqrt{2\sqrt2}}{2}$
$a_3 = \sqrt{\dfrac{a_2}{2}}=\sqrt{\dfrac{\frac{\sqrt{2\sqrt2}}{2}}{2}}=\sqrt{\dfrac{\sqrt{2\sqrt2}}{4}}=\dfrac{\sqrt{\sqrt{2\sqrt2}}}{2}=\dfrac{\sqrt[4]{2\sqrt2}}{2}$
$a_4 = \sqrt{\dfrac{a_3}{2}}=\sqrt{\dfrac{\frac{\sqrt[4]{2\sqrt2}}{2}}{2}}=\sqrt{\dfrac{\sqrt[4]{2\sqrt2}}{4}}=\dfrac{\sqrt{\sqrt[4]{2\sqrt2}}}{2}=\dfrac{\sqrt[8]{2\sqrt2}}{2}$
$a_5= \sqrt{\dfrac{a_4}{2}}=\sqrt{\dfrac{\frac{\sqrt[8]{2\sqrt2}}{2}}{2}}=\sqrt{\dfrac{\sqrt[8]{2\sqrt2}}{4}}=\dfrac{\sqrt{\sqrt[8]{2\sqrt2}}}{2}=\dfrac{\sqrt[16]{2\sqrt2}}{2}$
