Bonsoir,
1)
[tex] u_0=5\\
u_{n+1}=5u_n-6\\
w_n=u_n-\dfrac{3}{2} \\
S_n=\sum_{i=0}^n\ u_i\\\\
w_{n+1}=u_{n+1}-\dfrac{3}{2} \\
=5u_n-6-\dfrac{3}{2}\\
=5u_n-\dfrac{15}{2}\\
=5(u_n- \frac{3}{2} )\\
=5*w_n\\\\
w_0=\dfrac{7}{2}\\\\
\Rightarrow\ w_n=\dfrac{7}{2}*5^n\\
\Rightarrow\ u_n=\dfrac{7}{2}*5^n-\dfrac{3}{2}\\
S_n=\sum_{i=0}^n\ u_i\\
=\sum_{i=0}^n\ (\dfrac{7}{2}*5^i-\dfrac{3}{2} )\\
=\dfrac{7}{2}*\sum_{i=0}^n\ (5^i)\ -\dfrac{3}{2} *(n+1)\\
\\
[/tex]
[tex] =\dfrac{7}{2}*\dfrac{5^{n+1}-1}{4} -\dfrac{3}{2} *(n+1)\\
=\dfrac{7*5^{n+1}-12n-19}{8}
[/tex]
