Let
be an integer greater than
. Define
![x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i =...](/media/m/8/0/2/80297c16d4034850b8005614863d0012.png)
where
![[z]](/media/m/1/b/e/1be90bc55017944743736332d25924af.png)
denotes the largest integer less than or equal to
. Prove that
![\min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]](/media/m/a/4/e/a4e6b4231171a1d8c8a4bfba6f520b36.png)
%V0
Let $n$ be an integer greater than $1$. Define
$$x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i =...$$
where $[z]$ denotes the largest integer less than or equal to $z$. Prove that
$$\min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]$$
Školjka