Odpowiedź:
zad 1
a)
a₁ = - 27 , a₂ = 9 , a₃ = - 3
q = a₃/a₂ = a₂/a₁= -3/9 = 9/(- 27) = - 1/3 = - 1/3
b)
a₁ = 1/16 , a₂ = 1/4, a₃ = 1
q = a₃/a₂ = a₂/a₁= 1/(1/4) = (1/4)/(1/16) = 1 * 4 = 1/4 * 16 = 4 = 4
c)
a₁ = 2/9 , a₂ = 1/3 , a₃ = 1/2
q = a₃/a₂ = a₂/a₁= 1/2 : 1/3 = 1/3 : 2/9 = 1/2 * 3 = 1/3 * 9/2 = 3/2 = 3/2
zad 2
a)
a₁ =25 ,q = 1/5 ,n = 4
an = a₁ * qⁿ⁻¹ = 25 * (1/5)⁴⁻¹ =25 * (1/5)³ = 25 * 1/125 = 1 * 1/5 = 1/5
b)
a₁=1/10 , q = 5/2 = 2,5 , n = 4
an = a₁ * qⁿ⁻¹ = 1/10 * 2,5⁴⁻¹ = 1/10 * 2,5³ = 1/10 * 15,625 = 1,5625 =
= 1 5625/10000 = 1 225/400 = 1 9/16
zad 3
a)
a₁ = 1 , q = 2 , n = 8
S₈ = a₁(1 - q⁸)/(1 - q) = (1 - 2⁸)/(1 - 2) = (1 - 256)/( - 1) = - 255/(- 1) =
255/1 = 255
b)
a₁ = 15 , q = - 2 , n = 4
S₄ = a₁[(1 - (- 2)⁴]/(1 + 2) = 15(1 - 16)/3 = [15 * (- 15)]/3 = - 225/3 = - 75
zad 4
a)
a₁ = 2 , r = 2
an = a₁ + (n - 1) * r
a₆ = a₁ + (6 - 1) * r = 2 + 5 * 2 = 2 + 10 = 12
a₉ = a₁ + (9 - 1) * r = 2 + 8 * 2 = 2 + 16 = 18
b)
a₁ = - 8 , r = - 1/2
a₆ = a₁ + (6 - 1) * r = - 8 + 5 * (- 1/2) = - 8 - 5/2 = - 8 - 2 1/2 = - 10 1/2
