The rows $x_n$ and $y_n$ of positive real numbers are such that:
$x_{n+1}=x_n+\frac{1}{2y_n}$ and $y_{n+1}=y_n+\frac{1}{2x_n}$
for each positive integer $n$.
Prove that at least one of the numbers $x_{2018}$ and $y_{2018}$ is bigger than 44,9
Pinko wrote:
The rows $x_n$ and $y_n$ of positive real numbers are such that:
$x_{n+1}=x_n+\frac{1}{2y_n}$ and $y_{n+1}=y_n+\frac{1}{2x_n}$
for each positive integer $n$.
Prove that at least one of the numbers $x_{2018}$ and $y_{2018}$ is bigger than 44,9
We have $$x_{n+1}y_{n+1}=x_ny_n + \frac{1}{4x_ny_n}+1 > x_ny_n + 1.$$Assume that $x_{2018}\geq y_{2018}$, then $$x_{2018}^2\geq x_{2018}y_{2018} > x_1y_1+2017 > 2017$$and $x_{2018} > \sqrt{2017} > 44{,}9$.