Bonsoir,
1)
[tex]u_0\\u_{n+1}=a_n*u_n+b\\\\v_n=u_n-\dfrac{b}{1-a} \\\\v_{n+1}=u_{n+1}-\dfrac{b}{1-a} \\\\=a_n*u_n+b-\dfrac{b}{1-a}\\\\=a_n*u_n+\dfrac{b-ab-b}{1-a}\\\\=a(u_n-\dfrac{b}{1-a})\\\\=a*v_n\\[/tex]
2)
[tex]u_0=5,\ u_{n+1}=0.9*u_n+4\\v_n=u_n-\dfrac{4}{1-0.9} \\v_n=u_n-40\\v_0=u_0-40=5-40=-35\\\\v_n=-35*0.9^n\\\\u_n=40-35*0.9^n\\\\\\ \lim_{n \to \infty} u_n=40-0=40\\[/tex]
