Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

Question

Smart Achievers · Accepted Answer

(3)

$Let the numbers be a, a r, a r^{2}, we have$

$2$ $| 2 a r |$ $= a + a r^{2} \Rightarrow 4 r = r^{2} + 1 \Rightarrow r^{2} - 4 r + 1 = 0$

$\Rightarrow r = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2 \sqrt{3}}{2} = 2 \pm \sqrt{3}$

$∴$ $r = 2 + \sqrt{3}, as G.P. is increasing.$

Solution

Q.
Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

1 $3 + \sqrt{2}$

2 $2 - \sqrt{3}$

3 $2 + \sqrt{3}$

4 $\sqrt{2} + \sqrt{3}$

Ans.
(3)

$Let the numbers be a, a r, a r^{2}, we have$

$2$ $| 2 a r |$ $= a + a r^{2} \Rightarrow 4 r = r^{2} + 1 \Rightarrow r^{2} - 4 r + 1 = 0$

$\Rightarrow r = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2 \sqrt{3}}{2} = 2 \pm \sqrt{3}$

$∴$ $r = 2 + \sqrt{3}, as G.P. is increasing.$

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Solution

Q. Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

1 3+2

2 2-3

3 2+3

4 2+3

Ans. (3) Let the numbers be a, ar, ar2, we have 2|2ar|=a+ar2⇒4r=r2+1⇒r2-4r+1=0 ⇒ r=4±122=4±232=2±3 ∴ r=2+3, as G.P. is increasing.