iran-konkur 2020 Q115

iran-konkur · Other · konkur-riazi_1399_specialized_old-curriculum 4 marks Sequences and series, recurrence and convergence Convergence proof and limit determination
115. The sequence $\{x_n\}$ is defined as follows. What is the limit of $\{x_n\}$?
$$x_0 = 3 \;,\quad x_{n+1} = \frac{3x_n^2 + 64}{4x_n^2} \;,\quad (n = 1, 2, \ldots)$$
$$2\sqrt{2} \quad (1) \qquad -2\sqrt{2} \quad (2) \qquad 2\sqrt[4]{2} \quad (3) \qquad -2\sqrt[4]{2} \quad (4)$$
\textbf{115.} The sequence $\{x_n\}$ is defined as follows. What is the limit of $\{x_n\}$?

$$x_0 = 3 \;,\quad x_{n+1} = \frac{3x_n^2 + 64}{4x_n^2} \;,\quad (n = 1, 2, \ldots)$$

$$2\sqrt{2} \quad (1) \qquad -2\sqrt{2} \quad (2) \qquad 2\sqrt[4]{2} \quad (3) \qquad -2\sqrt[4]{2} \quad (4)$$