સમાગુણોતર શ્રેણીનું \(4\)મું પદ \(500\) છે અને તેનો સામાન્ય ગુણોતર \(\frac{1}{m}, m \in N\) છે.ધારોકે આ સમગુણોતર શ્રેણીના પ્રથમ \(n\) પદના સરવાળાને \(S_n\) વડે દર્શાવાય છે.જો \(S_6 > S_5+1\) અને \(S_7 < S_6+\frac{1}{2}\) હોય,તો \(m\)ની શક્ય કિંમતોની સંખ્યા \(.........\) છે.

A \(11\)
B \(10\)
C \(12\)
D \(15\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(12\)

Step-by-step Solution

Detailed explanation

\(T_4=500 \quad\) where \(a=\) first term, \(r =\) common ratio \(=\frac{1}{ m }, m \in N\) \(a r^3=500\) \(\frac{a}{m^3}=500\) \(S_n-S_{n-1}=a r^{n-1}\) \(S _6 > S _5+1 \quad\) and \(S _7- S _6 < \frac{1}{2}\) \(S _6- S _5 > 1 \quad \frac{ a }{ m ^6} < \frac{1}{2}\)…

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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