Question

the sum of three numbers in a.p.is15. if1 14 19 are addaed to the numbers the resulting number are in gp. find the numbers.

Answer 1

Sum of three numbers in A.P. is 15.

Now, let the three numbers be,

a-d , a , a+d.

Sum of the numbers is equal to 15.

Therefore,

a-d+a+a+d = 15

3a = 15

a=5.

Now,

the three numbers in A.P. are, 5-d,5,5+d

Adding 1,14,19 to the numbers will make them g.p which is infact if a,b&c are in g.p. then b^2=a*c

adding 1,14&19,

6-d,5+4,(5+d+19)

i.e., 6-d,9,24+d are in g.p where b^2=a*c i.e. b= sqr root(a*c)

9*9=(6-d)*(24+d)

144+6d-24d-d^2=81

144-18d-d^2=81

-18d-d^2=-63

d^2+18d-63=0

(d-3)*(d+21)=0

d=3,-21.

Taking d as 3,

the three numbers are a-d,a,a+d = 2,5,8.