Question

Three numbers form an increasing G.P. If the middle term is doubled, then the new
numbers are in A.P. Find the common ratio of the G.P.
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Answer 1

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Answer 2

Answer:

2 ± √3

Step-by-step explanation:

Let the three numbers be a,ar,ar².

Given that middle term of the G.P is doubled, then numbers are in A.P

a, 2ar, ar² are in A.P

⇒ 2(4ar) = a + ar²

⇒ 4ar = a + ar²

⇒ 4ar = a(1 + r²)

⇒ 4r = 1 + r²

⇒ r² - 4r + 1 = 0

On comparing with ax² + bx + c = 0, we get

a = 1, b = -4, c = 1

∴ D = b² - 4ac

      = (-4)² - 4(1)(1)

      = 12

The solutions are:

⇒ x = -b ± √D/2a

      = -(-4) ± √12/2

      = (4 ± √12)/2

      = 2 ± √3

Since the series is increasing.So,

⇒ r = 2 ± √3

Hope it helps!