Question -
Answer -
Let the first term ofan A.P. be ‘a’ and its common difference be‘d’.
b = a + d; c = a + 2d.
Given:
a + b + c = 18
3a + 3d = 18 or a + d= 6.
d = 6 – a … (i)
Now, according to thequestion:
a + 4, a + d + 4, anda + 2d + 36
they are now in GP,that is:
(a+d+4)/(a+4) =(a+2d+36)/(a+d+4)
(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
d = 6 –a
d = 6 – (– 2) or d = 6– 46
d = 8 or – 40
Then,
For a = -2 and d = 8,the A.P is -2, 6, 14
For a = 46 and d =-40, the A.P is 46, 6, -34
∴ The numbers are– 2, 6, 14 or 46, 6, – 34