Answer :

Let the original numbers be


a, a + d, and a + 2d


According to the question –


3a + 3d = 21 or a + d = 7.


d = 7 – a


After the addition, the three numbers are:


a, a + d – 1, and a + 2d + 1


they are now in GP, that is –



(a + d – 1)2 = a(a + 2d + 1)


a2 + d2 + 1 + 2ad – 2d – 2a = a2 + a + 2da


(7 – a)2 – 3a + 1 – 2(7 – a) = 0


49 + a2 – 14a – 3a + 1 – 14 + 2a = 0


a2 – 15a + 36 = 0


a2 – 12a – 3a + 36 = 0


a(a – 12) – 3(a – 12) = 0


a = 3 or a = 12


d = 7 – a


d = 7 – 3 or d = 7 – 12


d = 4 or – 5


The numbers are 3,7,11 or 12,7,2


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