SOLUTION: The 1st term of a g.p is x and the 3rd is y find an expression for the common ratio in terms of x and y

Algebra.Com
Question 1208200: The 1st term of a g.p is x and the 3rd is y find an expression for the common ratio in terms of x and y


Found 3 solutions by josgarithmetic, mananth, ikleyn:
Answer by josgarithmetic(39618)   (Show Source): You can put this solution on YOUR website!
If r is common ratio then first three terms can be as .

Last term is also .





Answer by mananth(16946)   (Show Source): You can put this solution on YOUR website!
The 1st term of a g.p is x and the 3rd is y find an expression for the common ratio in terms of x and y
The general form of a GP = a,ar, ar^2, ar^3
where r is the common ratio
a is the first term
lets replace a with x
x,xr, xr^2, xr^3.......
xr^2=y
r^2=y/x
r = (+-sqrt(y/x))







Answer by ikleyn(52792)   (Show Source): You can put this solution on YOUR website!
.

If 1st term of a GP is x and 3rd term is y, then

    y = ,    (1)


where r is the common ratio.  If x is not zero, then we can divide both sides of this equation by x

     = .    (2)


Notice that in this case, when x =/= 0,  from formula (1),  both  y and x have the same sign
(both are either positive or negative), so the ratio    is non-negative.


Hence,  we can take square root of  

    r = +/- ,


and in the case x =/= 0  it is the final answer to the problem's question, 
giving two possible values for r, if y =/= 0, or single zero, of y=0.



In the special case  x=0  we CAN NOT divide by x (by zero), so formula (2) does not work.


In this case, the geometric progression is degenerated: all its terms are zeros.


So, the complete answer is  r = +/- ,  if x =/= 0  (two possible values, if y =/= 0; or single 0, if y=0)
                                
                            or any real number, if x = 0, but the GP in this case is degenerated and consists of zeros.

Solved completely with giving a complete answer.

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

This problem is, obviously, a trap.
The trap is that a COMPLETE answer is required, including the case x=0.



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