3 numbers add to 27 

x + y + z = 27

and multiply together to make 288. 

xyz = 288


These numbers also fall into an arithmetic sequence.

That just means that they are equally spaced.  That is,
the diffence between the 2nd and 1st equals the difference
between the 3rd and 2nd.

y - x = z - y

which simplifies to

-x + 2y - z = 0 

So we have the system:

 x +  y + z = 27
xyz = 288
-x + 2y - z = 0

Adding the 1st and 3rd equations term by term

 x +  y + z = 27
-x + 2y - z =  0
----------------
     3y     = 27
          y = 9

Substitute that in the first and 2nd equations:

 x +  9 + z = 27          x(9)z = 288
      x + z = 18             xz = 32

So now we have the system

          x + z = 18             
             xz = 32

Solve the first for z:   z = 18 - x 
Substitute in the second equation:

      x(18 - x) = 32

       18x - x² = 32

 -x² + 18x - 32 = 0

  x² - 18x + 32 = 0

(x - 16)(x - 2) = 0

x - 16 = 0;  x - 2 = 0
     x = 16;     x = 2

Using x = 16      Using x = 2

z = 18 - x        z = 18 - x      
z = 18 - 16       z = 18 - 2
z = 2             z = 16

So x and z can switch places.  One is 16 and the other is 2

So the arithmetic sequence can either go:

2, 9, 16  where the common difference is 7

or

16, 9, 2  where the common difference is -7

Edwin