SOLUTION: The first term of a G.P is x and 3rd term is y. Find an expression for the common ratio in terms of x and y. the 1st and 3rd term of a G.P are 8 and 18 respectively and the com

Algebra ->  Sequences-and-series -> SOLUTION: The first term of a G.P is x and 3rd term is y. Find an expression for the common ratio in terms of x and y. the 1st and 3rd term of a G.P are 8 and 18 respectively and the com      Log On


   

Question 1189826: The first term of a G.P is x and 3rd term is y. Find an expression
for the common ratio in terms of x and y. the 1st and 3rd term of
a G.P are 8 and 18 respectively and the common ratio is positive,
find the common ratio.
Found 2 solutions by ikleyn, Edwin McCravy:
Answer by ikleyn(53942) About Me  (Show Source):
You can put this solution on YOUR website!
.
(a) the first term of a G.P is x and 3rd term is y. find an expression for the common ratio in terms of x and y.
(b) the 1st and 3rd term of a G.P are 8 and 18 respectively and the common ratio is positive, find the common ratio
~~~~~~~~~~~~~~~~~


                        Part (a) 


    a%5B1%5D = x;   a%5B3%5D = y = a%5B1%5D%2Ar%5E2.


    So,  y = x*r^2,  which gives  for the common ratio  r = +/- sqrt%28y%2Fx%29.      ANSWER



    Notice that this formula for the common ratio "r" works in real numbers  IF and only IF  

    the ratio  y%2Fx is non-negative, which means that both x and y must be of the same sign, and both are not zero.


                        Part (b) 


    a%5B1%5D = 8;   a%5B3%5D = 18 = a%5B1%5D%2Ar%5E2 = 8%2Ar%5E2.


    It gives  r = sqrt%2818%2F8%29 = sqrt%289%2F4%29 = +/- 3%2F2.      


    Since the condition says that common ratio is positive, 

    it leaves only one option for the common ratio  r = 3%2F2 = 1.5.   ANSWER

Solved.



Answer by Edwin McCravy(20086) About Me  (Show Source):
You can put this solution on YOUR website!
the first term of a G.P is x and 3rd term is y.
Find an expression for the common ratio in terms ]
of x and y.
So the sequence is like this
__x__, ___, __y__,...
r=matrix%281%2C2%2C2nd%2Cterm%29%2Fmatrix%281%2C2%2C1st%2Cterm%29%22%22=%22%22r=matrix%281%2C2%2C3rd%2Cterm%29%2Fmatrix%281%2C2%2C2nd%2Cterm%29%22%22=%22%22x%2Ar%22%22=%22%22y%2Fr
x%2Ar%5E2%22%22=%22%22y
r%5E2%22%22=%22%22y%2Fx
r%22%22=%22%22%22%22+%2B-+sqrt%28y%2Fx%29
The 1st and 3rd term of a G.P are 8 and 18 respectively
and the common ratio is positive, find the common ratio.
Plug in x=8 and y=18 in that and solve for r, using only 
the + of ±.

Edwin