SOLUTION: A geometric series has a second term of 6 and the sum of its first three terms is -14. Find it's fourth term. How do you work this out?

Algebra ->  Sequences-and-series -> SOLUTION: A geometric series has a second term of 6 and the sum of its first three terms is -14. Find it's fourth term. How do you work this out?      Log On


   

Question 356239: A geometric series has a second term of 6 and the sum of its first three terms is -14. Find it's fourth term. How do you work this out?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
a%5Bn%5D=a%5B1%5D%2Ar%5E%28n-1%29
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a%5B1%5D=a%5B1%5D%2Ar%5E%280%29=a%5B1%5D
a%5B2%5D=a%5B1%5D%2Ar=6
a%5B3%5D=a%5B1%5D%2Ar%5E2=6r
We can also look at a%5B1%5D in terms of a%5B2%5D.
a%5B1%5D=a%5B2%5D%2Fr=6%2Fr
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Now using the sum,
a%5B1%5D%2Ba%5B2%5D%2Ba%5B3%5D=-14
6%2Fr%2B6%2B6r=-14
6%2B6r%2B6r%5E2=-14r
6r%5E2%2B20r%2B6=0
3r%5E2%2B10r%2B3=0
%283r%2B1%29%28r%2B3%29=0
Two solutions:
3r%2B1=0
3r=-1
r=-1%2F3
a%5B1%5D=6%2Fr=-18
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r%2B3=0
r=-3
a%5B1%5D=6%2F%28-3%29=-2
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For the first solution, a%5B1%5D=-18, r=-1%2F3
a%5B4%5D=-18%2A%28-1%2F3%29%5E%283%29
highlight%28+a%5B4%5D=2%2F3%29
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For the second solution, a%5B1%5D=-2, r=-3
a%5B4%5D=-2%2A%28-3%29%5E%283%29
highlight%28+a%5B4%5D=54%29