Odpowiedź:
zad 1
a₈ = a₁q⁷ = 1
a₅ = a₁q⁴ = 27
a₈/a₅ = a₁q⁷/a₁q⁴ = q⁷⁻⁴ = q³
q³ = 1/27
q = ∛(1/27) = 1/3
a₁q⁴ = 27
a₁ = 27 : q⁴ = 27 : (1/3)⁴ = 27 : 81 = 27/81 = 1/3
an = a₁qⁿ⁻¹ = 1/3 * (1/3)ⁿ⁻¹
zad 2
a₁ = 48
a₂ = a₁q = x
a₃ = a₁q² = 3
a₃/a₂ = a₂/a₁
3/x = x/48
x² = 3 * 48 = 144
x² - 144 = 0
(x - 12)(x + 12) = 0
x - 12 = 0 ∨ x + 12 = 0
x = 12 ∨ x = - 12
zad 3
a₁ = 32
a₂ = a₁q = 16
a₂/a₁ = g
q = 16 : 32 = 1/2
a₃ = a₂q = 16 * 1/2 = 8
a₄ = a₃q = 8 * 1/2 = 4
a₅ = a₄q = 4 * 1/2 = 2
a₆ = a₅q = 2 * 1/2 =1
a₇ = a₆q = 1 * 1/2 = 1/2
a₈ = a₇q = 1/2 * 1/2 = 1/4
S₈ = a₁ + a₂ + a₃ + a₄ + a₅ + a₆ + a₇ + a₈ =32 + 16 + 8 + 4 + 2 + 1 + 1/2 + 1/4 =
= 63 + 1/2 + 1/4 = 63 + 2/4 + 1/4 = 63 3/4 = 63,75
II sposób
S₈ = [a₁(1 - q⁸)]/(1 - q) = [32(1 - (1/2)⁸)]/(1 - 1/2) = [32(1 - 1/256)/(1/2) =
= (32 * 255/256)/(1/2) = 8160/256 * 2 = 31,875 * 2 = 63,75
