Question 165
A sequence $(x_n)$ is defined as follows
$$
x_1 = 2, \quad x_{n+1} = \sqrt{x_n + 8} - \sqrt{x_n + 3}
$$
for all positive integers $n$.
a) Prove that $(x_n)$ has a finite limit and find that limit.
b) For every positive integer $n$, prove that
$$
n \le x_1 + x_2 + \dots + x_n \le n + 1.
$$
$$
x_1 = 2, \quad x_{n+1} = \sqrt{x_n + 8} - \sqrt{x_n + 3}
$$
for all positive integers $n$.
a) Prove that $(x_n)$ has a finite limit and find that limit.
b) For every positive integer $n$, prove that
$$
n \le x_1 + x_2 + \dots + x_n \le n + 1.
$$