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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.


Solution

Let the first term of the A.P. be a and the common difference be d.

a = a, b = a + d and c = a + 2d

a + b + c = 18

a+(a+d)+(a+2d)=18

3a+3d=18

a+d=6       ........(i)

Now, according to the question, a + 4, a + d + 4 and a + 2d + 36 are in G.P.

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

(6d+d+4)2=(6d+4)(6d+2d+36)

(6d+d+4)2=(6d+4)(6d+2d+36)

(10)2=(10d)(42+d)

100=420+10d42dd2

d2+32d320=0

(d+40)(d8)=0

d=8, 40

Now, putting d = 8, -40 in equation (i), we get, a = -2, 46, respectively.

For a = - 2, and d = 8, we have:

a = -2, b = 6, c = 14

And, for a = 46 and d = - 40, we have;

a = 46, b = 6, c = -34

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