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# 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. - Mathematics

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.

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#### 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

$a + b + c = 18$

$\Rightarrow a + \left( a + d \right) + \left( a + 2d \right) = 18$

$\Rightarrow 3a + 3d = 18$

$\Rightarrow a + d = 6 . . . . . . . (i)$

$\text { Now, according to the question, a + 4, a + d + 4 and a + 2d + 36 are in G . P .}$

$\therefore \left( a + d + 4 \right)^2 = \left( a + 4 \right)\left( a + 2d + 36 \right)$

$\Rightarrow \left( 6 - d + d + 4 \right)^2 = \left( 6 - d + 4 \right) \left( 6 - d + 2d + 36 \right)$

$\Rightarrow \left( 10 \right)^2 = \left( 10 - d \right)\left( 42 + d \right)$

$\Rightarrow 100 = 420 + 10d - 42d - d^2$

$\Rightarrow d^2 + 32d - 320 = 0$

$\Rightarrow \left( d + 40 \right)\left( d - 8 \right) = 0$

$\Rightarrow d = 8, - 40$

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

$\text { For a = - 2 and d = 8, we have }:$

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

$\text { And, for a = 46 and d = - 40, we have }:$

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

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 20 Geometric Progression
Exercise 20.5 | Q 6 | Page 45
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