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

$square root of 2 plus square root of 3$
$3 plus square root of 2$
$2 minus square root of 3$
$2 plus square root of 3$

Solution

2 plus square root of 3

Let a, ar, ar² be in GP (r>1)
On multiplying middle term by 2, a 2ar are in AP.
⇒ 4ar = a+ ar²
⇒ r²-4r +1 = 0
$straight r space equals space fraction numerator 4 plus-or-minus square root of 16 minus 4 end root over denominator 2 end fraction space equals space 2 plus-or-minus 3 straight r space equals space 2 plus square root of 3 space left square bracket space therefore space AP space is space increasing right square bracket$

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Permutations And Combinations

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

$square root of 2 plus square root of 3$
$3 plus square root of 2$
$2 minus square root of 3$
$2 plus square root of 3$

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