Let be in GP. Then are

in HP. Since are in GP, taking the natural logarithm gives in AP. Adding to each term keeps them in AP, so are in AP. The reciprocals of terms in an AP form an HP, so are in HP.

neither in AP nor in GP nor in HP

Let x \gt 1, y \gt 1, z \gt 1 be in GP. Then \frac{1}{1+\ln x}, \frac{1}{1+\ln y}, \frac{1}{1+\ln z} are

A. in AP
B. in GP
C. in HP ✓
D. neither in AP nor in GP nor in HP

Correct Answer: C. in HP

Explanation

Since x, y, z are in GP, taking the natural logarithm gives \ln x, \ln y, \ln z in AP. Adding 1 to each term keeps them in AP, so 1+\ln x, 1+\ln y, 1+\ln z are in AP. The reciprocals of terms in an AP form an HP, so \frac{1}{1+\ln x}, \frac{1}{1+\ln y}, \frac{1}{1+\ln z} are in HP.

Let x \gt 1, y \gt 1, z \gt 1 be in GP. Then \frac{1}{1+\ln x}, \frac{1}{1+\ln y}, \frac{1}{1+\ln z} are

Explanation

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