Assume 11 = ar^x

            12 = ar^y

            13 = ar^z  for x, y, z integers

       

Then

  12/11=r^(y-x), so ln(12/11)=(y-x)*ln(r)

  Similarly,              ln(13/12)=(z-y)*ln(r)

  

  Therefore,

  ln(12/11)/ln(13/12) = (y-x)/(z-y)

  

But the left hand side is irrational, and the right hand side is rational, so we have a contradiction.  Therefore, 11, 12 and 13 cannot be part of the same geometric progression.