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Question -

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.



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

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