Question

The 1st term of a G.P is equal to x-2,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.

Answer 1

arithmetic mean of 1st and 3d
= (x-2 + x+6)/2
= (2x + 4)/2
= x+2

let the 2nd be y
then y/(x-2) = (x+6)/y
y^2 = (x-2)(x-6)
y = ± √((x-2)(x+6))

mean:2nd = 5:3
(x+2)/√((x-2)(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)(x-3) = 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

Answer 2

Why are the arithmetic mean divided by 2