SOLUTION: The ratio of the sum of the first three terms of a G.P to the sum of the first 6 terms is 27:19. The common ratio is 2/3 -2/3 -1.5 1.5

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: The ratio of the sum of the first three terms of a G.P to the sum of the first 6 terms is 27:19. The common ratio is 2/3 -2/3 -1.5 1.5      Log On


   



Question 1135894: The ratio of the sum of the first three terms of a G.P to the sum of the first 6 terms is 27:19. The common ratio is
2/3
-2/3
-1.5
1.5

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
  a%5B1%5D + a%5B2%5D + a%5B3%5D + a%5B4%5D + a%5B5%5D + a%5B6%5D = 


= a+%2B+ar+%2B+ar%5E2+%2B+ar%5E3+%2B+ar%5E4+%2B+ar%5E5 = %28a+%2B+ar+%2B+ar%5E2%29 + r%5E3%2A%28a%2Bar+%2B+ar%5E2%29 = %281+%2B+r%5E3%29%2A%28a+%2B+ar+%2B+ar%5E2%29.


Therefore,


%28a%2Bar+%2B+ar%5E2%29%2F%28a+%2B+ar+%2B+ar%5E2+%2B+ar%5E3+%2B+ar%5E4+%2B+ar%5E5%29 = 1%2F%281+%2B+r%5E3%29 = 27%2F19.


Hence,   1+%2B+r%5E3 = 19%2F27   and  r%5E3 = 19%2F27+-+1 = 8%2F27.


It implies   r = root%283%2C8%2F27%29 = 2%2F3.


Answer.  r = 2%2F3,  which is the  first line of the option's list.