SOLUTION: 1.given that the tenth, fourth and firt terms of an A.P are the three consecutive terms of the G.P.find the common ratio and the sum of the first six terms,taking the first term to

Algebra ->  Sequences-and-series -> SOLUTION: 1.given that the tenth, fourth and firt terms of an A.P are the three consecutive terms of the G.P.find the common ratio and the sum of the first six terms,taking the first term to      Log On


   

Question 1006281: 1.given that the tenth, fourth and firt terms of an A.P are the three consecutive terms of the G.P.find the common ratio and the sum of the first six terms,taking the first term to be 4.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
We use a%5Bn%5D=a%5B1%5D%2B%28n-1%29d, for the nth term of an A.P.

>>1.given that the tenth, fourth and first terms of an A.P<< 
a%5B10%5D=a%5B1%5D%2B%2810-1%29d=4%2B9d

a%5B4%5D=a%5B1%5D%2B%284-1%29d=4%2B3d

a%5B1%5D=4

>>are the three consecutive terms of the G.P.<<
%284%2B9d%29r+=+4%2B3d or 4r%2B9dr=4%2B3d

%284%2B3d%29r+=+4    or 4r%2B3dr=4

We have the system:

system%284r%2B9dr=4%2B3d%2C4r%2B3dr=4%29

Solve that system by solving one for a variable and
substituting in the other.  (Lot of messy work). 
The two solutions are:

(d,r) = (0,1) or (4/3, 1/2)

The first solution, (d,r)=(0,1) gives the trivial sequence 
4,4,4,4,4,4,4... for both the A.P. with d=0 and a G.P. with r=1.

4, 4, 4, 4, 4, 4, 4, 4, 4, 4 = A.P. with common difference 0
4,       4,                4 = G.P. which common ratio 1...

The second solution (d,r) = (4/3, 1/2) gives the A.P

4, 16/3, 20/3, 8, 28/3, 32/3, 12, 40/3, 44/3, 16 = A.P.
4,             8,                           , 16 = G.P.

The sum of the first 6 terms is given by the sum formula:

S%5Bn%5D=expr%28n%2F2%29%282a%5B1%5D%2B%28n-1%29d%5E%22%22%29

S%5B6%5D=expr%286%2F2%29%282%2A4%2B%286-1%29%284%2F3%29%5E%22%22%29

S%5B6%5D=3%288%2B5%284%2F3%29%5E%22%22%29

S%5B6%5D=3%288%2B20%2F3%29

S%5B6%5D=24%2B20

S%5B6%5D=44

------------------

In the trivial case, 4,4,4,4,4,4,..., the sum of the first
6 terms is 24.

Edwin