Sum of three number a.p. is 15 if 1,4,19,... Are added respectively their form g.p. find the number.?

Let (a - d), a, (a + d) be the three numbers.

Sum of the three numbers :
(a - d) + a + (a + d) = 15
a = 5
Hence, the three numbers are (5 - d), 5 and (5 + d).

If 1, 4 and 19 are added respectively, they form g.p. Common ratio:
(5 + 4) / [(5 - d) + 1] = [(5 + d) + 19] / (5 + 4)
9 / (6 - d) = (24 + d) / 9
(6 - d)(24 + d) = 81
144 - 18d - d² = 81
d² + 18d - 63 = 0
(d - 3)(d + 21) = 0
d = 3 or d = -21

The three numbers are 2, 5, 8 or 26, 5, -16

The a.p. numbers are a, a+d and a+2d.
The g.p. numbers are x=a+1, y=a+d+4, and z=a+2d+19.
Use y/x = z/y to solve.
Hmmm, how DO you solve that?

By inspection, I was able to find a solution
2 5 8 has common difference 3
Add 1 4 19 and you get
3 9 27 which has common ratio 3.

But there should be an algebraic way to find that. I'm still working on that.
Oops! It would have helped if I had read the part where the sum of the 3 is 15.

wanszeto nailed it.

Let (a - d), a, and (a + d) be the three numbers.
Sum of the three numbers :
3a = 15
a = 5
Hence, the three numbers are (5 - d), 5 and (5 + d).
If 1, 4 and 19 are added respectively, they form g.p.
Common ratio:
(5 + 4) / [(5 - d) + 1] = [(5 + d) + 19] / (5 + 4)
9 / (6 - d) = (24 + d) / 9
(6 - d)(24 + d) = 81
d² + 18d - 63 = 0
(d + 21)(d - 3) = 0
d = -21 or d = 3
The three numbers are 2, 5, 8 or 26, 5, -16