Bonjour,
z₀ = 0 et zn+1 = λzn + i
1) z₁ = λz₀ + i = i
z₂ = λz₁ + i = λi + i = (λ + 1)i
z₃ = λz₂ + i = λ(λ + 1)i + i = (λ(λ + 1) + 1)i
2) Par récurrence :
n = 0 ⇒ (λ⁰ - 1)i/(λ - 1) = 0 = z₀
Hypothèse : Au rang n, zn = (λⁿ - 1)i/(λ - 1)
au rang n+1 : zn+1 = λzn + i
⇔ zn+1 = λ(λⁿ - 1)i/(λ - 1) + i
⇔ zn+1 = [λ(λⁿ - 1) + (λ - 1)]i/(λ - 1)
⇔ zn+1 = (λⁿ⁺¹ - 1)i/(λ - 1)
donc propriété héréditaire.
3) λ = i
a) z⁴ = (i⁴ - 1)i/(i - 1) = 0 (i⁴ = 1)
b) zn+4 = izn+3 + i
= i(zn+3 + 1)
= i(izn+2 + i + 1)
= -zn+2 - 1 + i
= -(izn+1 + i) - 1 + i
= -i(zn+1 + 1 - 1) - 1
= -izn+1 - 1
= -i(izn + i) - 1
= zn + 1 - 1
= zn
c)...
4) λ = 1 + i
z₀ = 0, z₁ = i,
z₂ = (λ + 1)i = (2 + i)i = -1 + 2i
z₃ = (λ(λ + 1) + 1)i = [(1 + i)(2 + i) + 1]i = (2 + 3i)i = -3 + 2i
z₄ = (1 + i)(-3 + 2i) + i = -5
5) bon, bah c'est tjs pareil...
