Question 73
Let $N$ be a positive integer. The sequence $x_1, x_2, \dots$ of non-negative reals is defined by
$$
x_n^2 = \sum_{i=1}^{n-1} \sqrt{x_i x_{n-i}}
$$
for all positive integers $n \gt N$. Show that there exists a constant $c \gt 0$, such that $x_n \le \frac{n}{2} + c$ for all positive integers $n$.
$$
x_n^2 = \sum_{i=1}^{n-1} \sqrt{x_i x_{n-i}}
$$
for all positive integers $n \gt N$. Show that there exists a constant $c \gt 0$, such that $x_n \le \frac{n}{2} + c$ for all positive integers $n$.