Answer :

Let d be the common difference of AP


b = a + d ; c = a + 2d.


Given: a + b + c = 18


3a + 3d = 18 or a + d = 6.


d = 6 – a


After the addition, the three numbers are:


a + 4, a + d + 4, and a + 2d + 36


they are now in GP, that is –



(a + d + 4)2 = (a + 2d + 36)(a + 4)


a2 + d2 + 16 + 8a + 2ad + 8d = a2 + 4a + 2da + 36a + 144 + 8d


d2 – 32a – 128


(6 – a)2 – 32a – 128 = 0


36 + a2 – 12a – 32a – 128 = 0


a2 – 44a – 92 = 0


a2 – 46a + 2a – 92 = 0


a(a – 46) + 2(a – 46) = 0


a = – 2 or a = 46


As,


d = 6 –a


d = 6 – ( – 2) or d = 6 – 46


d = 8 or – 40


numbers are – 2, 6, 14 or 46, 6, – 34


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