Question

three numbers whose sum is 15 are in A,P.;if 1,4,19 are added to them respectively,result are in G.P.determine the numbers

Answer 1

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

Given that their sum is 15
⇒(a-d) + a + (a+d) = 15
⇒3a = 15
⇒a = 15/3 = 5

If 1,4,19 are added respectively, the numbers would become (a-d+1), (a+4) and (a+d+19) or (6-d), 9 and (24+d). Since these are in GP

$\frac{9}{6-d} = \frac{24+d}{9}$

⇒$(6-d)(24+d) = 9*9$

⇒$144-18d- d^{2} =81$

⇒$63-18d- d^{2} = 0$

⇒ $d^{2}+18d-63= 0$

⇒ $d = 3$

5-3 = 2 and 5+3 = 8

So the numbers are 2,5,8