Réponse :
Explications étape par étape :
Bonjour,
[tex]\left\{\begin{array}{ccc}u_6=7\\\\u_{n+1}=\dfrac{9}{10} *u_n\\\\\displaystyle v_n=\sum_{k=6}^n (u_k)\end {array} \right.\\\\\\u_6=7=u_0*(\dfrac{9}{10})^6 \\u_0=\dfrac{7}{(\dfrac{9}{10})^6} \\u_n=u_0*(\dfrac{9}{10})^n =\dfrac{7}{(\dfrac{9}{10})^6} *(\dfrac{9}{10})^n\\\\\\\boxed{u_n=7*(\dfrac{9}{10})^{n-6}}\\\\\\[/tex]
[tex]\displaystyle v_n=\sum_{k=6}^n (u_k)=\sum_{k=6}^n (7*(\dfrac{9}{10})^{k-6})\\\\=7*\sum_{i=0}^{n-6} ((\dfrac{9}{10})^{i})\\\\\\v_n=7*\dfrac{(\dfrac{9}{10})^{n-5}-1}{\dfrac{9}{10}-1} \\\\\\\boxed{v_n=70*(1- (\dfrac{9}{10})^{n-5} ) }\\\\[/tex]
