ત્રણ ધન સંખ્યાઓ વધતી સમગુણોતર શ્રેણી બનાવે છે. જો આ સમગુણોતર શ્રેણીના મધ્યમ પદને બમણું કરવામાં આવે તો નવી સંખ્યાઓ સંમાતર શ્રેણીમાં થાય. તો સમગુણોતર શ્રેણીનો સામાન્ય ગુણોતર ........ થાય.

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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