If `a ,b ,c` are three distinct real numbers in G.P. and `a+b+c=x b ,` then prove that either x < -1 or x > 3

Explanation

`a+b+c = xb`

`=>xb-b = a+c`

`=>b(x-1) = a+c`

`=>x-1 = (a+c)/b`

As, `a,b,c` are in G.P,

`:. b^2 = ac => b = sqrt(ac).`
`:. x-1 = (a+c)/(sqrt(ac))`

`=>x -1 = sqrt(a/c)+sqrt(c/a)`

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