Bonsoir,
Pas évident ce truc!
[tex]\boxed {u_{0}=1} \\
\boxed{u_{n+1}= 2* u_{n} -n+1=2* u_{n} +2 -(n+1)}\\
\boxed{u_{n}= 2* u_{n-1} +2 -n}\\
u_{0}=1\\
u_{1}=2*1+2-1=3\\
u_{2}=2*3+2-2=6\\
u_{3}=2*6+2-3=11\\
u_{4}=2*11+2-4=20\\
u_{5}=2*20+2-5=37\\
avec\\
u_{1}-1=2=2^1\\
u_{2}-2=4=2^2\\
u_{3}-3=8=2^3\\
u_{4}-4=16=2^4\\
u_{5}-5=32=2^5\\
[/tex]
[tex]\boxed{u_{n}-n=2^n}\\
D\'emonstration \\
u_{0}-0=2^0=1\\
\mbox{on suppose la propri\'et\'e vraie pour n}\\
u_{n}-n=2^n\\
u_{n+1}= 2* u_{n} +2 -(n+1)\\
u_{n+1}-(n+1)= 2* u_{n} +2 -2(n+1)\\
= 2*(u_{n} +1 -n-1)\\
= 2*(2^n)=2^{n+1}\\
[/tex]
[tex]S_{n} = u_{0} +u_{1} +u_{2} +....+u_{n} \\
S_{n}-(0+1+2+3+...+n)=2^0+2^1+2^2+...+2^n= \dfrac{2^{n+1}-1}{2-1} \\
S_{n}=2^{n+1}-1+ \dfrac{n*(n+1)}{2} \\
[/tex]
