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Question

Five numbers are in HP. The middle term is $$1$$ and the ratio of the second and the fourth terms is $$2:1$$. Then, the sum of the first three terms is


A
112
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B
5
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C
2
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D
143
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Solution

The correct option is A $$\cfrac{11}{2}$$
Let $$a-2d,a-d,a+d,a+2d$$ are in AP
then $$\cfrac{1}{a-2d},\cfrac{1}{a-d},\cfrac{1}{a+d},\cfrac{1}{a+2d}$$ are in HP
Given, middle term $$=1$$
$$\Rightarrow$$ $$\cfrac{1}{a}=1$$ 
and $$\cfrac{\text{Second term}}{\text{First term}}=\cfrac{2}{1}$$
$$\Rightarrow$$ $$\cfrac{1}{a-d}\times (a+d)=\cfrac{2}{1}$$
$$\Rightarrow$$ $$a=3d$$ $$\Rightarrow$$ $$d=\cfrac{1}{3}$$  ($$\because a=1$$)
$$\therefore$$ Sum of first three terms
$$=\cfrac{1}{1-\cfrac{2}{3}}+\cfrac{1}{1-\cfrac{1}{3}}+1=3+\cfrac{3}{2}+1=\cfrac{11}{2}$$

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