Answer :

Let the original numbers be


a, a + d, and a + 2d


According to the question –


a + a + d + a + 2d = 15


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


d = 5 – a


After the addition, the three numbers are:


a + 1, a + d + 3, and a + 2d + 9


they are now in GP, that is –



(a + d + 3)2 = (a + 2d + 9)(a + 1)


a2 + d2 + 9 + 2ad + 6d + 6a = a2 + a + 2da + 2d + 9a + 9


(5 – a)2 – 4a + 4(5 – a) = 0


25 + a2 – 10a – 4a + 20 – 4a = 0


a2 – 18a + 45 = 0


a2 – 15a – 3a + 45 = 0


a(a – 15) – 3(a – 15) = 0


a = 3 or a = 15


d = 5 – a


d = 5 – 3 or d = 5 – 15


d = 2 or – 10


The numbers are 3,5,7 or 15,5, – 5


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