Odpowiedź:
zad 1
an = - 6/n + 3
a₃ = - 6/3 + 3 = - 2 + 3 = 1
a₁₈ = - 6/18 + 3 = - 1/3 + 3 = 2 2/3
zad 2
a₄ = a₁ + 3r = 3
a₁₆ = a₁ + 15r = - 3
a)
Układ równań
a₁ + 3r = 3
a₁ + 15r = - 3
odejmujemy równania
a₁ - a₁ + 3r - 15r = 3 + 3
- 12r = 6
12r = - 6
r = - 6/12 = - 1/2
a₁ + 3r = 3
a₁ + 3 * (- 1/2) = 3
a₁ - 3/2 = 3
a₁ = 3 + 3/2 = 3 + 1 1/2 = 4 1/2 = 4,5
b)
a₃₀ = a₁ + 29r = 4 + 29 * (- 1/3) = 4 - 29/3 = 4 - 9 1/3 = - 5 1/3
S₃₀ = (a₁ + a₃₀) * 30/2 = (4 - 5 1/3) * 15 = - 1 1/3 * 15 = - 4/3 * 15 =
= - 4 * 5 = - 20
zad 3
a₁ = 6
a₂ = 2x + 1
a₃ = 54
a₃/a₂ = a₂/a₁
a₂² = a₁ * a³
(2x + 1)² = 6 * 54
4x² + 4x + 1 = 324
4x² + 4x + 1 - 324 = 0
4x² + 4x - 323 = 0
a = 4 , b = 4 , c = - 323
Δ = b² - 4ac = 4² - 4 * 4 * (- 323) = 16 + 5168 = 5184
√Δ = √5184 = 72
x₁ = (- b - √Δ)/2a = (- 4 - 72)/8 = - 76/8 = - 9 4/8 = - 9 1/2 = - 9,5
x₂ = (- b + √Δ)/2a = (- 4 + 72)/8 = 68/8 = 8 4/8 = 8 1/2 = 8,5
Ponieważ ciąg jest rosnący więc x = 8,5
a₁ = 6
a₂ = 2x + 1 = 2 * 8,5 + 1 = 17 + 1 = 18
a₃ = 54
sprawdzenie
a₂² = a₁ * a₃
18² = 6 * 54
324 = 324
L = P
