666. Выпишите первые пять членов геометрической прогрессии (bₙ), заданной формулой п-го члена, и найдите их сумму: а) b = 6 ⋅ (2/3)^(n-1); б) b = -81 ⋅ (2/3)^(n-1)

Решение:

а) bₙ = 6 ⋅ (2/3)^(n-1)

Вычисляем первые пять членов:

• b₁ = 6 ⋅ (2/3)⁰ = 6 ⋅ 1 = 6
• b₂ = 6 ⋅ (2/3)¹ = 6 ⋅ (2/3) = 4
• b₃ = 6 ⋅ (2/3)² = 6 ⋅ (4/9) = 8/3
• b₄ = 6 ⋅ (2/3)³ = 6 ⋅ (8/27) = 16/9
• b₅ = 6 ⋅ (2/3)⁴ = 6 ⋅ (16/81) = 32/27

Сумма первых пяти членов:

\[ S_5 = 6 + 4 + \frac{8}{3} + \frac{16}{9} + \frac{32}{27} = \frac{162 + 108 + 72 + 48 + 32}{27} = \frac{422}{27} = 15\frac{17}{27} \]

б) bₙ = -81 ⋅ (2/3)^(n-1)

Вычисляем первые пять членов:

• b₁ = -81 ⋅ (2/3)⁰ = -81 ⋅ 1 = -81
• b₂ = -81 ⋅ (2/3)¹ = -81 ⋅ (2/3) = -54
• b₃ = -81 ⋅ (2/3)² = -81 ⋅ (4/9) = -36
• b₄ = -81 ⋅ (2/3)³ = -81 ⋅ (8/27) = -24
• b₅ = -81 ⋅ (2/3)⁴ = -81 ⋅ (16/81) = -16

Сумма первых пяти членов:

\[ S_5 = -81 - 54 - 36 - 24 - 16 = -211 \]