Question

Three numbers form an increasing GP. If the middle term is doubled, than the new
numbers are in AP. The common ratio of GP is
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Answer 1

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Three numbers form an increasing GP. If the middle term is doubled, than the new

numbers are in AP. The common ratio of GP is?

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$\bf \: 2 - \sqrt{3}$

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Let the numbers be a, ar and ar² in GP where r is common ratio.

According to the question,

a, 2ar and ar² are in AP then

4ar = a + ar²

4r = 1 + r²

r² - 4r + 1 = 0

$\bf \: r = \frac{ - ( - 4) + \sqrt{( - 4) {}^{2} - 4 \times 1 \times 1} }{2 \times 1}$

$\bf \: r = \frac{4 + \sqrt{16 - 4} }{2}$

$\bf \: r = \frac{4 + \sqrt{12} }{2}$

$\bf \: r = \frac{4 + 2 \sqrt{3} }{2}$

$\bf \: r = 2 + \sqrt{3.}$

Hence, this is the value of common ratio of given GP.