Math.G.P
posted by Miracle .
The 1st term of a G.P is equal to x2,the 3rd term is x+6 and the arithmetic mean of the 1st and 3rd terms stands in the ratio to the 2nd 5:3.find the value of x.

arithmetic mean of 1st and 3d
= (x2 + x+6)/2
= (2x + 4)/2
= x+2
let the 2nd be y
then y/(x2) = (x+6)/y
y^2 = (x2)(x6)
y = ± √((x2)(x+6))
mean:2nd = 5:3
(x+2)/√((x2)(x+6)) = 5/3
square both sides and expand
(x^2 + 4x + 4)/(x^2 + 4x  12) = 25/9
25x^2 + 100x  300 = 9x^2 + 36x + 36
16x^2 + 64x  336 = 0
x^2 + 4x  21 = 0
(x+7)(x3) = 0
x = 7 or x = 3
check:
if x = 7,
the 3 terms are : 9, ±3, 1
(only the sequence 9, 3, 1 works)
if x = 3
the 3 terms are :1, ±3, 9
(only the sequence 1 , 3, 9 works)
so x = 7 or x = 3