2. Given a sequence of positive terms $\left\{a_{n}\right\}$ satisfying:
$$
a_{1}=\frac{1}{2}, a_{n+1}^{2}=\frac{2 a_{n}^{2}}{1+2 a_{n}^{2}} \text {. }
$$

then $a_{n}=$
$\begin{array}{l}\text { 2. } \sqrt{\frac{2^{n-2}}{1+2^{n-1}}} \text {. } \\ \text { From } \frac{1}{a_{n+1}^{2}}=\frac{1+2 a_{n}^{2}}{2 a_{n}^{2}}=1+\frac{1}{2} \cdot \frac{1}{a_{n}^{2}} \\ \Rightarrow \frac{1}{a_{n+1}^{2}}-2=\frac{1}{2}\left(\frac{1}{a_{n}^{2}}-2\right) \\ \Rightarrow \frac{1}{a_{n}^{2}}=\left(\frac{1}{a_{1}^{2}}-2\right)\left(\frac{1}{2}\right)^{n-1}+2 \\ \Rightarrow a_{n}=\sqrt{\frac{2^{n-2}}{1+2^{n-1}}} .\end{array}$