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Question

 Find the three numbers a,b,c between 2 and 18 such that
 (I) their sum in 25 
(II) the numbers 2, a, b are consecutive terms of an AP 
(III) the numbers b,c,18 are consecutive terms of a GP.


A
5,8,12
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B
5,8,10
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C
8,10,12
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D
4,5,8
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Solution

The correct option is A 5,8,12
Given $$a,b,c$$ are three numbers between 2 and 18 and $$ a+b+c=25$$ ...(i)
2,a,b are consecutive terms of A.P, then
so we can write , $$a\;=\;\frac{(b\;+\;2)}{2}$$...(ii)
also given b,c,18 are consecutive terms of a GP
so it implies $$c^2\;=\;18b$$...(iii)
From equations (i), (ii) and (iii), we get 
$$b=8 $$ and $$32$$ , since $$b$$ is between 2 and 18 , therefore taking $$ b = 8 $$
and then taking the value of b and putting it in other 2 equations we get the values of $$a\;=\;5$$  and  $$c\;=\;12$$


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