Bonjour Gabanna
[tex]\left\{\begin{matrix}u_0=-4\ \ \ \ \ \ \ \\u_{n+1}=u_n^2-u_n+1 \end{matrix}\right.\\\\\\1)\ u_{1}=u_0^2-u_0+1=(-4)^2-(-4)+1=16+4+1=\boxed{21} \\\\u_{2}=u_1^2-u_1+1=21^2-21+1=441-21+1=\boxed{421} \\\\u_{3}=u_2^2-u_2+1=421^2-421+1=\boxed{176821}\\\\u_{4}=u_3^2-u_3+1=176821^2-176821+1\approx\boxed{3,1\times10^{10}}[/tex]
2) Sens de variation de la suite (un)
[tex]u_{n+1}-u_n=(u_n^2-u_n+1) - u_n\\\\u_{n+1}-u_n=u_n^2-2u_n+1\\\\u_{n+1}-u_n=(u_n^2-1)^2\ge0\\\\\boxed{u_{n+1}\ge u_n}[/tex]
Par conséquent, la suite est croissante.
