SUBIECTUL IV

Consider the sequence:

$\left(x_{n}\right)_{n \geq 1}$ of real numbers defined by $x_{1}=\sqrt{\frac{1}{2}}, \quad x_{n+1}=\sqrt{\frac{1+x_{n}}{2}}, \quad n \geq 1$. Calculate $\lim _{n \rightarrow \infty} x_{1} \cdot x_{2} \cdot \ldots . . x_{n}$

## BAREM CLASA XI
## SUBJECT IV

$$
\begin{aligned}
& x_{1}=\cos \frac{\pi}{4}(1 p) ; x_{2}=\cos \frac{\pi}{8}(1 p) ; x_{n}=\cos \frac{\pi}{2^{n+1}}(1 p) ; x_{1} \cdot x_{2} \cdot \ldots \ldots \cdot x_{n}=\frac{1}{2^{n} \sin \frac{\pi}{2^{n+1}}}(2 p) ; \\
& \lim _{n \rightarrow \infty} x_{1} \cdot x_{2} \cdot \ldots . . \cdot x_{n}=\frac{2}{\pi}(2 p)
\end{aligned}
$$