Question 83
Let be given a real number $\alpha \neq 0$. Consider the sequence of real numbers $\{x_n\}$, $n = 1, 2, 3, \dots$, defined by:
$$
x_1 = 0 \quad \text{and} \quad x_{n+1}(x_n + \alpha) = \alpha + 1 \quad \text{for every } n = 1, 2, 3, \dots
$$
1/ Find the general term of the sequence $\{x_n\}$.
2/ Prove that the sequence $\{x_n\}$ has a finite limit when $n \to +\infty$. Find this limit.
$$
x_1 = 0 \quad \text{and} \quad x_{n+1}(x_n + \alpha) = \alpha + 1 \quad \text{for every } n = 1, 2, 3, \dots
$$
1/ Find the general term of the sequence $\{x_n\}$.
2/ Prove that the sequence $\{x_n\}$ has a finite limit when $n \to +\infty$. Find this limit.