teyannaharris9155 teyannaharris9155
  • 25-01-2024
  • Mathematics
contestada

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is 27/19. Then the common ratio of this series is :
A 4/9
B 2/9
C 2/3
D 1/3

Respuesta :

andyzhuangzixian andyzhuangzixian
  • 25-01-2024

Answer:   B) 2/9

Step-by-step explanation:

Sum = a / (1 - r)

3 = a / (1 - r)   ----(1)

Sum of cubes = a^3 / (1 - r^3)

27/19 = a^3 / (1 - r^3)   ----(2)

a = 3(1 - r)

27/19 = (3(1 - r))^3 / (1 - r^3)

27/19 = (27(1 - r)^3) / (1 - r^3)

27/19 = (1 - r)^3 / (1 - r^3)

27(1 - r^3) = 19(1 - r)^3

27 - 27r^3 = 19 - 57r + 57r^2 - 57r^3

30r^3 + 57r^2 - 57r + 8 = 0

Sum = 3 / (1 - 2/9) = 3 / (7/9) = 27/7

Sum of cubes = (2/9)^3 / (1 - (2/9)^3) = 8/729 / (1 - 8/729) = 8/729 / (721/729) = 8/721

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