Odpowiedź:
zad 1
a)
an = - 5 - (1 - 2n)/3
- 5 - (1 - 2n)/3 = - 2 | * 3
- 5 * 3 - 1 + 2n = - 2 * 3
- 15 - 1 + 2n = - 6
- 16 + 2n = - 6
2n = - 6 + 16
2n = 10
n = 10/2 = 5
Odp: a₅ = - 2
b)
- 5 - (1 - 2n)/3 < 0 | * 3
- 5 * 3 - 1 + 2n < 0
- 15 - 1 + 2n < 0
- 16 + 2n < 0
2n < 16
n < 8
Ponieważ n ∈ N⁺ więc 7 wyrazów tego ciągu jest mniejszych od 0
zad 2
an = (- 3n + 2)/5
a₁ = (- 3 * 1 + 2)/5 = (- 3 + 2)/5 = - 1/5
a₂ = (- 3 * 2 + 2)/5 = (- 6 + 2)/5 = - 4/5
a₃ = (- 3 * 3 + 2)/5 = ( - 9 + 2)/5 = - 7/5
a₃ - a₂ = a₂ - a₁
- 7/5 + 4/5 = - 4/5 + 1/5
- 3/5 = - 3/5
L = P c.n.w
zad 3
a₁ = 59
a₂ = a₁ + r = 55
a₃ = a₂ + r = 59
r = a₂ - a₁ = 55 - 59 = - 4
a₄₀ = a₁ + 39r = 59 + 39 * (- 4) = 59 - 156 = - 97
S₄₀ = (a₁ + a₄₀) * 40/2 = (59 - 97) * 20 = - 38 * 20 = - 760
zad 4
a₆ = a₁ + 5r = 0
a₉ = a₁ + 8r = 6
Układ równań
a₁ + 5r = 0
a₁ + 8r = 6
Odejmujemy równania
a₁ + a₁ + 5r - 8r = 0 - 6
- 3r = - 6
3r = 6
r = 6/3 = 2
a₁ + 5r = 0
a₁ + 5 * 2 = 0
a₁ + 10 = 0
a₁ = - 10
Wzór ogólny
an = a₁ + (n - 1) * r = - 10 + (n - 1) * 2 = - 10 + 2n - 2 = 2n - 12
