तीन धनात्मक संख्याएं बढ़ती गुणोत्तर श्रेढ़ी में हैं। यदि इस गुणोत्तर श्रेढी की बीच वाली संख्या दुगुनी कर दो जाए, तो नई बनी संख्याएं समांतर श्रेढ़ी में हो जाती हैं। गुणोत्तर श्रेढ़ी का सार्वअनुपात है:

A \(2 - \sqrt 3 \)
B \(2 + \surd 3\)
C \(\sqrt 2 + \surd 3\)
D \(3 + \surd 2\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(2 + \surd 3\)

Step-by-step Solution

Detailed explanation

Let the numbers be \(a, a r, a r^{2}\) If \(a r\) is doubled then, \(\Longrightarrow 2 a r=\frac{a r^{2}+a}{2}\) \(\Longrightarrow 4 a r=a r^{2}+a\) \(\Longrightarrow 4 r=r^{2}+1\) \(\Longrightarrow r^{2}-4 r+1\) On solving we get \(r=2 \pm \sqrt{3}\) since it is an increasing…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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