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Question

# Three positive numbers form an increasing GP. If the middle term in this GP is doubled, then new numbers are in AP. Then, the common ratio of the GP is

A

$2+\sqrt{3}$

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B

$3+\sqrt{2}$

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C

$2-\sqrt{3}$

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D

$2+\sqrt{2}$

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Solution

## The correct option is A $2+\sqrt{3}$Explanation for the correct answer:Let the three terms of the geometric progression be $\frac{a}{r},a,ar$It is given that if the middle term is doubled then the resultant terms will form an arithmetic progression$â‡’$ $\frac{a}{r},2a,ar$ are in an arithmetic progression $â‡’$ $2a=\frac{\frac{a}{r}+ar}{2}$$â‡’$ $4a=\frac{a{r}^{2}+a}{r}$$â‡’$ $4ar=a{r}^{2}+a$$â‡’$ ${r}^{2}-4r+1=0$Using formula for quadratic equation we get,$r=\frac{4Â±\sqrt{{4}^{2}-4}}{2}$$â‡’$ $r=\frac{4Â±2\sqrt{3}}{2}$$â‡’$ $r=2Â±\sqrt{3}$As it is an increasing geometric progression $r$ cannot be less than $1$Hence, $r=2+\sqrt{3}$Hence, the common ratio of the geometric progression is $2+\sqrt{3}$Hence, option A is the correct answer.

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