Question 10
For each real number $x_{1}$, the sequence $x_{1}, x_{2}, \ldots, x_{n}, \ldots$ is constructed by making $$
x_{n+1} = x_{n}\left(x_{n} + \frac{1}{n}\right) \text{ for each } n \geq 1
$$ show that there is exactly one value of $x_{1}$ for which $0 \lt x_{n} \lt x_{n+1} \lt 1$ for each $n$.
x_{n+1} = x_{n}\left(x_{n} + \frac{1}{n}\right) \text{ for each } n \geq 1
$$ show that there is exactly one value of $x_{1}$ for which $0 \lt x_{n} \lt x_{n+1} \lt 1$ for each $n$.