Answer
$a_{n}=2^{(2^{n}-1)/2^{n}}$
Work Step by Step
We are given:
$\sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, \sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}$, ...
We write each terms as a power of $2$:
$a_1=\sqrt{2}=2^{1/2}$
$a_{2}=\sqrt{2^1*2^{1/2}}=\sqrt{2^{3/2}}=2^{3/4}$
$a_{3}=\sqrt{2^1*2^{3/4}}=\sqrt{2^{7/4}}=2^{7/8}$
$a_{4}=\sqrt{2^1*2^{7/8}}=\sqrt{2^{15/8}}=2^{15/16}$
We see that the denominator in the power doubles (powers of 2) and the numerator is the denominator minus 1.
Therefore:
$a_{n}=2^{(2^{n}-1)/2^{n}}$
