569. Выпишите первые пять членов последовательности (ₙ), если: a) a₁ = 1, an+1 = an + 1; б) а₁ = 1000, an+1 = 0,1an; в) а₁ = 16, an+1 = -0,5an; г) а₁ = 3, ап+1 = aₙ⁻¹.

Ответ: a) 1, 2, 3, 4, 5; б) 1000, 100, 10, 1, 0.1; в) 16, -8, 4, -2, 1; г) 3, 1/3, 3, 1/3, 3

Решение:

а) Дано: a₁ = 1, aₙ₊₁ = aₙ + 1

• a₂ = a₁ + 1 = 1 + 1 = 2
• a₃ = a₂ + 1 = 2 + 1 = 3
• a₄ = a₃ + 1 = 3 + 1 = 4
• a₅ = a₄ + 1 = 4 + 1 = 5

б) Дано: a₁ = 1000, aₙ₊₁ = 0,1aₙ

• a₂ = 0,1 ⋅ a₁ = 0,1 ⋅ 1000 = 100
• a₃ = 0,1 ⋅ a₂ = 0,1 ⋅ 100 = 10
• a₄ = 0,1 ⋅ a₃ = 0,1 ⋅ 10 = 1
• a₅ = 0,1 ⋅ a₄ = 0,1 ⋅ 1 = 0,1

в) Дано: a₁ = 16, aₙ₊₁ = -0,5aₙ

• a₂ = -0,5 ⋅ a₁ = -0,5 ⋅ 16 = -8
• a₃ = -0,5 ⋅ a₂ = -0,5 ⋅ (-8) = 4
• a₄ = -0,5 ⋅ a₃ = -0,5 ⋅ 4 = -2
• a₅ = -0,5 ⋅ a₄ = -0,5 ⋅ (-2) = 1

г) Дано: a₁ = 3, aₙ₊₁ = \(\frac{1}{a_n}\)

• a₂ = \(\frac{1}{a_1}\) = \(\frac{1}{3}\)
• a₃ = \(\frac{1}{a_2}\) = \(\frac{1}{\frac{1}{3}}\) = 3
• a₄ = \(\frac{1}{a_3}\) = \(\frac{1}{3}\)
• a₅ = \(\frac{1}{a_4}\) = \(\frac{1}{\frac{1}{3}}\) = 3

Ответ: a) 1, 2, 3, 4, 5; б) 1000, 100, 10, 1, 0.1; в) 16, -8, 4, -2, 1; г) 3, 1/3, 3, 1/3, 3