Bonjour Oceean
[tex]\left\{\begin{matrix}u_1=1\\u_{n+1}=f(u_n)=\dfrac{2u_n-1}{u_n+4}\end{matrix}\right.[/tex]
1) Calculs de u2, u3 et u4.
[tex]u_2=\dfrac{2u_1-1}{u_1+4}=\dfrac{2\times1-1}{1+4}=\dfrac{2-1}{1+4}=\dfrac{1}{5}\\\\u_3=\dfrac{2u_2-1}{u_2+4}=\dfrac{2\times\dfrac{1}{5}-1}{\dfrac{1}{5}+4}=\dfrac{\dfrac{2}{5}-\dfrac{5}{5}}{\dfrac{1}{5}+\dfrac{20}{5}}=\dfrac{-\dfrac{3}{5}}{\dfrac{21}{5}}=-\dfrac{3}{21}=-\dfrac{1}{7}\\\\u_4=\dfrac{2u_3-1}{u_3+4}=\dfrac{2\times(-\dfrac{1}{7})-1}{-\dfrac{1}{7}+4}=\dfrac{-\dfrac{2}{7}-\dfrac{7}{7}}{-\dfrac{1}{7}+\dfrac{28}{7}}=\dfrac{-\dfrac{9}{7}}{\dfrac{27}{7}}=-\dfrac{9}{27}=-\dfrac{1}{3}[/tex]
2) Liste des premiers termes.
[tex]u_2=\dfrac{1}{5}\\\\u_3=-\dfrac{1}{7}\\\\u_4=-\dfrac{1}{3}\\\\u_5=-\dfrac{5}{11}\\\\u_6=-\dfrac{7}{13}\\\\u_7=-\dfrac{3}{5}=-0,6\\\\u_8=-\dfrac{11}{17}\approx-0,647\\\\u_9=-\dfrac{13}{19}\approx-0,684\\...\\u_{30}=-\dfrac{55}{61}\approx-0,9016\\...\\u_{50}=-\dfrac{95}{101}\approx-0,94[/tex]
La suite (un) paraît décroissante et semble se rapprocher de la valeur -1.
[tex]3)\ u_{n+1}=\dfrac{2u_n-1}{u_n+4}\\\\u_{n+1}=\dfrac{2u_n+8-8-1}{u_n+4}\\\\u_{n+1}=\dfrac{(2u_n+8)-9}{u_n+4}\\\\u_{n+1}=\dfrac{2(u_n+4)-9}{u_n+4}\\\\u_{n+1}=\dfrac{2(u_n+4)}{u_n+4}-\dfrac{9}{u_n+4}\\\\\boxed{u_{n+1}=2-\dfrac{9}{u_n+4}}[/tex]
