72. Find, based on geometric considerations, the limit

$$
\lim _{n \rightarrow \infty} 2^{n} \sqrt{\underbrace{2-\sqrt{2+\sqrt{2+\sqrt{2+\ldots+V \sqrt{2}}}}}_{(n-1) \text { twos }}}
$$
72. The sides of regular inscribed polygons are expressed in terms of $R$ of the circumscribed circle as follows:

\[
\begin{aligned}
& a_{4}=R \sqrt{2}, \quad a_{8}=R \sqrt{2-\sqrt{2}}, \quad a_{16}=R \sqrt{2-\sqrt{2+\sqrt{2}}}, \ldots \\
& a_{2^{n+1}}=R \sqrt{2-\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}}
\end{aligned}
\]

Therefore, the perimeter of a regular $2^{n+1}$-gon is $2^{n+1} a_{2^{n+1}}$, hence, $\lim _{n \rightarrow \infty} 2^{n+1} a_{2^{n+1}}=2 \pi R$, and the desired limit is $\pi$.