arithmeticogeo.py

Created on October 16, 2023
1.09 KB
Un+1= 0,8Un+30
U0= 200
Vn= Un-150
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1) ni arithmetique ni geo
U0=...
U1=...
U2=...
  A: U1-U0=...
     U2-U1=...
(U1-U0)=/ (U2-U1) pas ari
  G: U1/U0=...
     U2/U1=...
(U1/U0)=/ (U2/U1) pas geo
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2)(Vn) est geo + q + 1er term
Vn= Un-150
Un+1= 0,8Un+30
Un= Vn+150

-> V0= 200-150 =50 <- 1er term
-> Vn+1= Un+1 -150
       = 0,8Un+30 -150
       = 0,8(Vn+150)+30 -150
       = 0,8(Vn+150)-120
       = 0,8Vn+120 -120
   Vn+1= 0,8Vn
   q=0,8
-> (Vn) est geo
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3)Vn enfct de n +Un enfct de n
->Vn=q^n xV0
    =0,8^n x50
->Un=Vn+150
    =0,8^n x50 +150
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4)variation de (Un)
0<0,8<1  <- decroissante

=0,8^(n+1) <0,8^(n)       ]x50
=0,8^(n+1)x50 <0,8^(n)x50 ]+150
=0,8^(n+1)x50+150<0,8^(n)x50+150

=> Un^(n+1) < Un 
(Un) est decroissant
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5)Calculer Tn=V0+V1...+Vn
puis Sn=U0+U1+...+Un

V0=50  Un=Vn+150  q=0,8
ax(b/c)=(a/c)xb
Tn=V0 x (1-q^(n+1))/(1-q)
Tn=...

Sn=U0+U1+...+Un
->U0=V0+150
 +U1=V1+150
  ...
 +Un=Vn+150
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Sn=Tn+150(n+1)