let the original numbers be x , 4x and 13x
(4x+1)/(x+1) = (13x+1)/(4x+1)
(4x+1)^2 = (x+1)(13x+1)
16x^2 + 8x + 1 = 13x^2 + 14x + 1
3x^2 - 6x = 0
3x(x - 2) = 0
x = 0 or x = 2
we'll reject the x = 0
so the 3 numbers are 2, 8 and 26
check: after adding 1 to each, the numbers would be 3, 9, and 27
which would be in a GP
If we accept 0,0,0, then add 1 to each, we get
1,1,1, which technically satisfy the definition of a GP, with r = 1
but in most definition for a GP, r ≠1
besides that, are 0,0,0 in the ratio of 1:4:13 ??
My conclusion is that the original numbers are 2, 8, and 26