On pose {\begin{cases}a_{0}=1\\b_{0}=1\end{cases}} et {\begin{cases}(1) :a_{n+1}=a_{n}\!+\!b_{n}/2\\(2) :b_{n+1}=r_{n}\left(4a_{n}\!+\!b_{n}\right)\end{cases}}

1. Justifier que {\lim\limits_{+\infty}a_{n}=\lim\limits_{+\infty}b_{n}=+\infty }.
2. On suppose que {n\mapsto q_{n}=\dfrac{a_{n+1}}{a_{n}}} est monotone. Donner sa limite, et {k>0} tel que {b_{n}\stackrel{+\infty}{\sim}ka_{n}}.