for the what value of x the numbers x-2, x,x+3 are in g.p.

Answer 1

when their ratios are equal. So, you need

x/(x-2) = (x+3)/x
x^2 = (x+3)(x-2)
x^2 = x^2+x-6
x-6 = 0
x = 6

4,6,9 have the common ratio 3/2