Răspuns:
Explicație pas cu pas:
a) x₁ = 0 ; xₙ₊₁ = 5xₙ+3 ; n ≥ 1
x₂ = 5x₁+3 = 5·0+3 = 3
x₃ = 5x₂+3 = 5·3+3 = 18
x₄ = 5x₃+3 = 5·18+3 = 93
b) x₁ = √5 ; xₙ₊₁ = √(5+xₙ)
x₂ = √(5+√5)
x₃ = √[5+√(5+√5)]
x₄ = √{5+√[5+√(5+√5)]}
c) x₁ = 2 ; xₙ = 1/2 ·xₙ₋₁ + 1 ; n ≥ 2
x₂ = 1/2 ·x₁ +1 = 1/2 ·2 + 1 = 1+1 = 2
x₃ = 1/2 ·x₂+1 = 1/2 ·2 + 1 = 1+1 = 2
x₄ = 1/2 ·x₃+1 = 1/2 ·2 + 1 = 1+1 = 2
d) x₁ = 0 ; xₙ - xₙ₋₁ = (1/2)ⁿ⁺¹ ; n ≥ 2
x₂ = x₁+(1/2)²⁺¹ = 0 + 1/2³ = 1/8
x₃ = x₂+(1/2)⁴ = 1/8 + 1/16 = 3/16
x₄ = x₃+(1/2)⁵ = 3/16+1/32 = 7/32
e) x₁ = 1/2 ; xₙ₊₁ = xₙ/(1+2xₙ) ; n≥ 1
x₂ = x₁ / (1+2x₁) = 1/2 / (1+1) = 1/4
x₃ = x₂/(1+2x₂) = 1/4 / (1+1/2) = 1/4 / 3/2 = 1/4 ·2/3 = 2/12 = 1/6
x₄ = x₃/(1+2x₃) = 1/6 / (1+1/3) = 1/6 / 4/3 = 1/6 · 3/4 = 3/24 = 1/8
f) x₁ = 3/2 ; xₙ = n / (n+2) ·xₙ₋₁ ; n ≥ 2
x₂ = 2/(2+2) ·3/2 = 1/2 ·3/2 = 3/4
x₃ = 3/(3+2) ·3/4 = 3/5 ·3/4 = 9/20
x₄ = 4(4+2) ·9/20 = 2/3 ·9/20 = 18/60 = 3/10
