Bonjour,
1) U₀ = 1/2
U₁ = 2U₀/(2 + 7U₀) = 1/(2 + 7/2) = 1/(11/2) = 2/11
U₂ = 2U₁/(2 + 7U₁) = (4/11)/(2 + 14/11) = (4/11)/(36/11) = 1/9
U₃ = 2U₂/(2 + 7U₂) = (1/9)/(2 + 7/9) = (1/9)/(25/9) = 1/25
U₁ - U₀ = 2/11 - 1/2 = (4 - 11)/22 = -7/22
U₂ - U₁ = 1/9 - 2/11 = (11 - 18)/99 = -7/99
Donc (U₂ - U₁) ≠ (U₁ - U₀) ⇒ (Un) n'est pas arithmétique
2) Vn = (2 - Un)/Un
V₀ = (2 - 1/2)/(1/2) = 3
V₁ = (2 - 2/11)/(2/11) = 10
V₂ = (2 - 1/9)(1/9) = 17
V₃ = (2 - 1/25)/(1/25) = 24
Vn+1 = (2 - Un+1)/Un+1 = 2/Un+1 - 1
= 2/[2Un/(2 + 7Un)] - 1
= (2 + 7Un)/Un - 1
= (2 + 7Un - Un)/Un
= (2 - Un)/Un + 7Un/Un
= Vn + 7
Donc (Vn) arithmétique de raison r = 7 et de premier terme V₀ = 3
3) On en déduit :
Vn = V₀ + 7n = 3 + 7n
et donc : (2 - Un)/Un = 3 + 7n
⇔ 2 - Un = (3 + 7n)Un
⇔ Un = 2/(4 + 7n)
4) U₁₀ = 2/(4 + 70) = 2/74 = 1/37
