Je vous prête le bonjour,
[tex]u_1=20\\
u_{n+1}= \dfrac{3*u_n}{2} +2\\\\
1)\\
u_2=\dfrac{3*20}{2} +2=32\\\\
u_3=\dfrac{3*32}{2} +2=50\\\\
2)\\
u_2-u_1=32-20=12\\
u_3-u_2=50-32=18\\ la\ suite\ n\'\ est\ pas\ arithm\'etique.\\\\
\dfrac{u_2}{u_1} = \dfrac{32}{20} =1.6\\
\dfrac{u_3}{u_2} = \dfrac{50}{32} =1.5625\ la \ suite \ n \' \ est \ pas \ g\'eom\'etrique.
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[/tex]
[tex]3)\\
v_n=u_n+4\\
a)\\
v_{n+1}=u_{n+1}+4\\
= \dfrac{3u_n}{2}+2+4= \dfrac{3}{2}*(u_n+4)=\dfrac{3}{2}*v_n\\\\
2*v_{n+1}-3v_n=0\\
b)\\
v_1=20+4=24\\
r=\dfrac{3}{2}\\
c)\\
v_n=24*( \frac{3}{2} )^{n-1}\\
v_9=24* (\frac{3}{2})^{8}=615,09375\approx{615.09}\\
[/tex]
[tex]4)\\
p_2=p_1+ \frac{p_1}{2} = \frac{3}{2} *p_1\\
p_3=p_2+ \frac{p_2}{2} = \frac{3}{2} *p_2\\
p_{n+1}=p_n+ \frac{p_n}{2} = \frac{3}{2} *p_n\\
[/tex]
[tex]5)\\
a)\\
p_n=p_1* (\frac{3}{2} )^{n-1}\\\\
=240* (\frac{3}{2} )^{n-1}\\
b)\\
s=p_1+p_2+p_3+p_4\\
=p_1*( (\frac{3}{2} )^{0}+(\frac{3}{2} )^{1}+(\frac{3}{2} )^{2}+(\frac{3}{2} )^{3})\\\\
=p_1*\dfrac{ (\frac{3}{2} )^{4}-1) } { \dfrac{3}{2}-1}\\
=480*( (\dfrac{3}{2} )^{4}-1) =1950\\\\
c)\\
p_n=240* (\frac{3}{2} )^{n-1} \ \textgreater \ 6150\\
(\dfrac{3}{2} )^{n-1}\ \textgreater \ \dfrac{205}{8}\\
n-1\ \textgreater \ \dfrac{ln(\dfrac{205}{8})}{ln(1.5)} \\
n\ \textgreater \ 8.99962406...\\
n=9\\
[/tex]
