Question 121
A natural number $m$ is given. The sequence $a_{n}$ is defined by the conditions $a_{1}=m$ and $a_{n+1} = \left[\frac{a_{n}^{2} + 10 a_{n}}{2 a_{n} + 4}\right]$ for all $n \in \mathbb{N}$. Prove that the sequence $a_{n}$ is constant from some term on.
Note. For any real number $x$, the notation $[x]$ denotes the greatest integer not greater than $x$.
Note. For any real number $x$, the notation $[x]$ denotes the greatest integer not greater than $x$.