The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.
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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