Algebra help please quickly?

Question

3 nos are in GP.sum of their squares is S^2.

their sum is xS.Find the range of x^2.

please help me quickly.

Answer 1

let the terms in the GP be    a, ar, ar^2

where a is the first term and r is the ratio

sum of squares = S^2

a^2 + a^2r^2 + a^2r^4 = S^2            --------e(1)

sum of 3 terms

a + ar + ar^2 = xS

squaring both sides

(a + ar + ar^2)^2 = x^2S^2           --------e(2)

e(2) /  e(1) we get

(a + ar + ar^2)^2    /   a^2 + a^2r^2 + a^2r^4   =   x^2S^2 / S^2

(a^2) * (1 + r + r^2)^2

--------------------------------------...            = x^2

(a^2) * (1 + r^2 + r^4)

(1 + r^2 + r^4 + 2r + 2r^2 + 2r^3)

------------------------------------- ----------------------------------------...            = x^2

(1+ r^2 + r^4)              

             2r ( 1 + r + r^2)

            -------------------------  +  1             =  x^2        --------e(3)

             (1+ r^2 + r^4)              

degree of numerator = 3 and degree of denominator = 4

The degree of numerator is less then degree of denominator, Hence the function magnitude converges to (x^2 = 1) as we go from r = 1 to infinity and from r = -1 to -infinity

The plot of the above function (with r on x axis and x^2 on y-axis) has a minimum at (-1,  1 / 3 ) and maximum at (1,3). check by substituting r = 1 and r = -1 in e(3)

Thus the range of x^2 is [1/3, 3]         ------> solved

we have also plotted the curve (r,x^2) see the following link

http://img510.imageshack.us/img510/5997/...

Answer 2

a, b and c

a^2+b^2+c^2=s^2

a+b+c=xs

x=(a+b+c)/s

x^2 = (a+b+c)^2/s^2

x^2 = (a+b+c)^2/a^2+b^2+c^2