WBJEE · Maths · Sequences and Series

If \(a, b, c\) are in G.P. \((a>1, b>1, c>1)\), then for any real number \(x(\) with \(x>0, x \neq 1), \log _{\mathrm{a}} x, \log _{\mathrm{b}} x, \log _{\mathrm{c}} x\) are in

A G.P.
B A.P.
C H.P.
D G.P. but not in H.P.

Verified Solution

Answer & Solution

Correct Answer

(C) H.P.

Step-by-step Solution

Detailed explanation

Hints : \(a, b, c\) are in G.P. \(\Rightarrow \log _x a, \log _x b, \log _x c\) are in A.P. \(\Rightarrow \frac{1}{\log _x a}, \frac{1}{\log _x b}, \frac{1}{\log _x c}\) arein H.P. \(\Rightarrow \log _a X, \log _b X, \log _c X\) are in H.P.

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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