Let the terms be, a, a+d, a+2d. Given that sum is 12.
That is, a+(a+d)+(a+2d) = 12
That is, 3a+3d = 12,
Since sum is 12, numbers can be (1,4,7) or (2,4,6) or (3,4,5).
But it's given that sum of cubes is 408. This is satisfied only by (1,4,7).
Hence the numbers are 1,4,7.