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.

Asked by Sakshi | 1 year ago |  51

##### Solution :-

Let the first term of an 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 the question:

a + 4, a + d + 4, and a + 2d + 36

they are now in GP, that is:

$$\dfrac{ (a+d+4)}{(a+4)}$$$$\dfrac{(a+2d+36)}{(a+d+4)}$$

(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

Answered by Aaryan | 1 year ago

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