A.P. (Arithmetic Progression)

Given:
- The sum of three positive numbers constituting an A.P. is 15.

Let the three numbers be a-d, a, and a+d, where "a" is the middle term and "d" is the common difference.

The sum of an A.P. can be calculated using the formula:
Sum = (n/2)(2a + (n-1)d)

Since we have three terms, n = 3.
Substituting the values in the formula, we get:
15 = (3/2)(2a + 2d)
15 = 3a + 3d

Simplifying the equation:
5 = a + d

G.P. (Geometric Progression)

After adding 1, 4, and 19 to the three numbers respectively, we get a G.P.

The terms of the G.P. can be represented as (a-d+1), (a+1), and (a+d+19).

The common ratio of a G.P. can be calculated using the formula:
Common ratio (r) = (second term)/(first term)

Substituting the values, we get:
r = (a+1)/(a-d+1)

Similarly, the third term divided by the second term should also give the same common ratio:
r = (a+d+19)/(a+1)

Equating the two expressions for the common ratio:
(a+1)/(a-d+1) = (a+d+19)/(a+1)

Simplifying the equation:
(a+1)^2 = (a-d+1)(a+d+19)
a^2 + 2a + 1 = a^2 + 19a + d^2 + 19d - a - d + 1
2a = 19a + d^2 + 19d - d
18a = d^2 + 19d

Substituting the value of a from the equation 5 = a + d:
18(5-d) = d^2 + 19d

Simplifying the equation:
90 - 18d = d^2 + 19d
d^2 + 37d - 90 = 0

Solving the quadratic equation, we get:
(d-2)(d+45) = 0
d = 2 or d = -45

Since we are considering positive numbers, d = 2.

Substituting the value of d in the equation 5 = a + d:
5 = a + 2
a = 3

Therefore, the three numbers are 1, 3, and 5.

No.s added are 1,4,18
so answer is 3,5,7