Question

The sum of 3numbers in AP is 15 . if 1,4,19 be added to them respectively, the result s in GP .The numbers are

Answer 1

Let the original numbers be 
a, a+d, and a+2d 

3a + 3d = 15 or a+d = 5 ----> d = 5-a 

after the addition, the three numbers are: 
a+1, a+d+4, and a+2d+19 
they are now in GP, that is .... 
(a+d+4)/(a+1) = (a+2d+19)/(a+d+4) 
(a + 5-a + 4)/(a+1) = (a + 10-2a + 19)/(a + 5-a + 4) 
9/(a+1) = (-a + 29)/9 
81 = -a^2 + 28a + 29 
a^2 - 28a + 52 = 0 
(a - 26)(a - 2) = 0 

a = 26 or a = 2 

if a = 26, then d = 5-26 = -21 
and the original 3 numbers were: 
26, 5, and 16 

Answer 2

(a-d)+a+(a+d) = 15 i.e. a=5.
Now, 5-d+1, 5+4, 5+d+19 are in GP.
That means (5-d+1)(5+d+19) = 9^2.
144-18d-d^2 = 81.
d^2+18d-63 = 0.
d = -21,3.

So the numbers are..
2,5,8.