Odpowiedź:
a₁ + a₂ + a₃ = 15
a₁ + a₁ + r + a₁ + 2r = 15
3a₁ + 3r = 15 | : 3
a₁ + r = 5 ⇒ a₁ = 5 - r
a₁ * a₂ * a₃ = 80
a₁ * (a₁ + r) * (a₁ + 2r) = 80
a₁(a₁² + a₁r + 2a₁r + 2r²) = 80
a₁(a₁² + 3a₁r + 2r²) = 80
(5 - r)[(5 - r)² + 3(5 - r) * r + 2r²] = 80
(5 - r)(25 - 10r + r² + 15r - 3r² + 2r²) = 80
(5 - r)(25 + 5r) = 80
5(5 - r)(5 + r) = 80
5(25 - r²) = 80
25 - r² = 80 : 5 = 16
- r² = 16 - 25 = - 9
r² = 9
r² - 9) = 0
(r - 3)(r + 3) = 0
r - 3 = 0 ∨ r + 3 = 0
r = 3 ∨ r = - 3
a₁ = 5 - r ∨ a₁ = 5 + r
a₁ = 5 - 3 ∨ a₁ = 5 + 3
a₁ = 2 ∨ a₁ = 8
Obliczamy wyrazy ciągu
1.
dla a₁ = 2 i r = 3
a₁ = 2
a₂ = a₁ + r = 2 + 3 = 5
a₃ = a₁ + 2r = 2 + 2 * 3 = 2 + 6 = 8
2.
dla a₁ = 8 i r - - 3
a₁ = 8
a₂ = a₁ + r = 8 - 3 = 5
a₃ = a₁ + 2r = 8 + 2 * (- 3) = 8 - 6 = 2
Dla tych wartości a₁ = 8 i r = - 3 ciąg jest ciągiem malejącym więc :
a₁ = 2 i r = 3
an = a₁ + (n - 1) * r = 2 + (n - 1) * 3 = 2 + 3n - 3 = 3n - 1
Odp:
a₁ = 2 , r = 3 , an = 3n - 1
