Question -
Answer -
Let the first term ofan A.P. be┬атАШaтАЩ┬аand its common difference beтАШdтАЩ.
b = a + d; c = a + 2d.
Given:
a + b + c = 18
3a + 3d = 18 or a + d= 6.
d = 6 тАУ a тАж (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┬а=┬а(a+ 2d + 36)(a + 4)
a2┬а+ d2┬а+16 + 8a + 2ad + 8d = a2┬а+ 4a + 2da + 36a + 144 + 8d
d2┬атАУ32a тАУ 128
(6 тАУ a)2┬атАУ32a тАУ 128 = 0
36 + a2┬атАУ12a тАУ 32a тАУ 128 = 0
a2┬атАУ44a тАУ 92 = 0
a2┬атАУ46a + 2a тАУ 92 = 0
a(a тАУ 46) + 2(a тАУ 46)= 0
a = тАУ 2 or a = 46
d = 6 тАУa
d = 6 тАУ (тАУ 2) or d = 6тАУ 46
d = 8 or тАУ 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
тИ┤┬аThe numbers areтАУ 2, 6, 14 or 46, 6, тАУ 34