SOLUTION: Can you explain the solution to this problem please? In a G.P. the sum of the 2nd and 3rd terms is 6, the sum of the 3rd and 4th terms is -12. Find the 1st term and the common r

Algebra ->  Sequences-and-series -> SOLUTION: Can you explain the solution to this problem please? In a G.P. the sum of the 2nd and 3rd terms is 6, the sum of the 3rd and 4th terms is -12. Find the 1st term and the common r      Log On


   

Question 648175: Can you explain the solution to this problem please?
In a G.P. the sum of the 2nd and 3rd terms is 6, the sum of the 3rd and 4th terms is -12. Find the 1st term and the common ratio.
Thanks very much
Answer by MathTherapy(10858) About Me  (Show Source):
You can put this solution on YOUR website!

Can you explain the solution to this problem please?
In a G.P. the sum of the 2nd and 3rd terms is 6, the sum of the 3rd and 4th terms is -12. Find the 1st term and the common ratio.
Thanks very much

GP formula: a%5Bn%5D+=+a%5B1%5Dr%5E%28n+-+1%29

a%5B2%5D+=+a%5B1%5Dr%5E%282+-+1%29 ----- a%5B2%5D+=+a%5B1%5Dr

a%5B3%5D+=+a%5B1%5Dr%5E%283+-+1%29 ----- a%5B3%5D+=+a%5B1%5Dr%5E2

a%5B4%5D+=+a%5B1%5Dr%5E%284+-+1%29 ----- a%5B4%5D+=+a%5B1%5Dr%5E3+

Since terms 2 and 3 sum to 6, then:
a%5B1%5Dr+%2B+a%5B1%5Dr%5E2+=+6 ---- a%5B1%5D%28r+%2B+r%5E2%29+=+6 ---- a%5B1%5D+=+6%2F%28r%5E2+%2B+r%29 ----- eq (i)

Since a%5B3%5D+=+6+-+a%5B1%5Dr, since a%5B4%5D+=+a%5B1%5Dr%5E3, and since a%5B3%5D+%2B+a%5B4%5D+=+-+12, then: 6+-+a%5B1%5Dr+%2B+a%5B1%5Dr%5E3+=+-+12 ------ eq (ii)

a%5B1%5Dr%5E3+-+a%5B1%5Dr+=+-+18 ------ eq (ii)
%286%2F%28r%5E2+%2B+r%29%29+%2A+r%5E3+-+%286%2F%28r%5E2+%2B+r%29%29+%2A+r+=+-+18 ------ Substituting 6%2F%28r%5E2+%2B+r%29 for a%5B1%5D in eq (ii)

%286%2F%28r%28r+%2B+1%29%29%29+%2A+r%5E3+-+%286%2F%28r%28r+%2B+1%29%29%29+%2A+r+=+-+18



%286r%5E2%29%2F%28r+%2B+1%29+-+6%2F%28r+%2B+1%29+=+-+18

6r%5E2+-+6+=+-+18%28r+%2B+1%29 ------ Multiplying by LCD, r + 1

6r%5E2+-+6+=+-+18r+-+18

6r%5E2+%2B+18r+-+6+%2B+18+=+0

6r%5E2+%2B+18r+%2B+12+=+0

6%28r%5E2+%2B+3r+%2B+2%29+=+2%280%29

r%5E2+%2B+3r+%2B+2+=+0

(r + 2)(r + 1) = 0

r = - 2 or – 1

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Check
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r = - 2
a%5B1%5Dr%5E2+%2B+a%5B1%5Dr+=+6

a%5B1%5D%28-+2%29%5E2+%2B+a%5B1%5D%28-+2%29+=+6

4a%5B1%5D+-+2a%5B1%5D+=+6

2a%5B1%5D+=+6

a%5B1%5D+=+3

highlight_green%28r+=+-+2%29; First term, or a%5B1%5D = highlight_green%283%29

r = - 1
a%5B1%5Dr%5E2+%2B+a%5B1%5Dr+=+6

a%5B1%5D%28-+1%29%5E2+%2B+a%5B1%5D%28-+1%29+=+6

a%5B1%5D+-+a%5B1%5D+%3C%3E+6

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