$$\begin{array}{l}
{a_1} = 0;{a_2} = 1; \hfill \\
{a_n} = \sqrt {\frac{{{a_{n - 1}} + {a_{n - 2}}}}{2} \cdot {a_{n - 1}}} ,{\text{ }}n \geqslant 3; \hfill \\
\mathop {\lim }\limits_{n \to \infty } {a_n} = \frac{1}{\pi }{\rm B}\left( {\frac{1}{2},\frac{3}{4}} \right) \hfill \\
\end{array} $$