Since the three numbers form an AP with the sum 18, the middle terms is 18/3 = 6.


Let "d" be the common difference of this AP.


Then the three terms of the AP are  6-d, 6 and 6+d.


Then the three terms of the GP are  ((6-d)+4) = 10-d, 6 and 6+d.


Since the terms  10-d, 6 and 6+d  form a GP,


     = .


It is your equation to find "d". Cross multiply


    6d = (10-d)*(6+d)

    6d = 60 - 6d + 10d - d^2

    d^2 + 2d - 60 = 0

     =  =  = .



With the middle term 6,  NEITHER  value of  d =  = 6.81...  NOR   d =  = -8.81...

provides three positive terms of the AP.



ANSWER.  As the problem is worded, printed, posted and presented, the solution DOES NOT exist.