ધારોકે \(a_1, a_2, a_3, \ldots\) એ વધતી પૂર્ણાંક સંખ્યાઓ ની સમગુણોતર શ્રેણી છે. જો ચોથા અને છઠા પદોનો ગુણાકાર \(9\) હોય અને સાતમુપદ \(24\) હોય, તો \(a_1 a_9+a_2 a_4 a_9+a_5+a_7=...................\)

A \(600\)
B \(606\)
C \(60\)
D \(6\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(60\)

Step-by-step Solution

Detailed explanation

\(a_4 \cdot a_6=9 \Rightarrow\left(a_5\right)^2=9 \Rightarrow a_5=3\) \(a_5+a_7=24 \Rightarrow a_5+a_5 r^2=24 \Rightarrow\left(1+r^2\right)=8 \Rightarrow r=\sqrt{7}\) \(\Rightarrow a=\frac{3}{49}\) \(\Rightarrow a_1 a_9+a_2 a_4 a_9+a_5+a_7=9+27+3+21=60\)

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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