Odpowiedź:
zad 3
a)
an = 1/2n² - 18 n ≥ 1
1/2n² - 18 = 0
1/2n² = 18
n² = 2 * 18 = 36
n = √36 = 6
sprawdzenie
a₆ = 1/2 * 6² - 18 = 1/2 * 36 -18 = 18 - 18 = 0
b)
an = (n² - 4n)/n + 1 n ≥ 1
n² - 4n = 0
n(n - 4) = 0
n = 0 odrzucamy ze względu na założenie n ≥ 1
n - 4 = 0
n = 4
sprawdzenie
a₄ = (4² - 4 * 4)/(4 + 1) = (16 - 16)/5 = 0/5 = 0
zad 4
a)
a₁ = 100 , a₂ = 98
r - różnica ciągu = a₂ - a₁ = 98 - 100 = - 2
a₆ = a₁ + 5r = 100 + 5 * (- 2) = 100 - 10 = 90
a₇ = a₆ + r = 90 + (- 2) = 90 - 2 = 88
b)
a₁ = - 14 , a₂ = - 12
r = a₂ - a₁ = - 12 - (- 14) = - 12 + 14 = 2
a₆ = a₁ + 5r = - 14 + 5 * 2 = - 14 + 10 = - 4
a₇ = a₆ + r = - 4 + 2 = - 2
zad 5
a)
an = 4n - 7
a₁ = 4 * 1 - 7 = 4 - 7 = - 3
a₂ = 4 * 2 - 7 = 8 - 7 = 1
a₃ = 4 * 3 - 7 = 12 - 7 = 5
a₃ - a₂ = a₂ - a₁
5 - 1 = 1 - (- 3)
5 - 1 = 1 + 3
4 = 4
L = P
Jest to ciąg arytmetyczny
b)
an = 1/2n + 3
a₁ = 1/2 * 1 + 3 = 1/2 + 3 = 3 1/2
a₂ = 1/2 * 2 + 3 = 1 + 3 = 4
a₃ = 1/2 * 3 + 3 = 3/2 + 3 = 1 1/2 + 3 = 4 1/2
a₃ - a₂ = a₂ - a₁
4 1/2 - 4 = 4 - 3 1/2
1/2 = 1/2
L = P
Jest ro ciąg arytmetyczny
