Let x >1, y >1, z >1 be in GP. Then 1/1+ln x, 1/1+lny, 1/1+ z are

neither in AP nor in GP nor in HP

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UPSC NDA II 2024 Mathematics Paper

Question : 11 of 120

Marks: +1, -0

\rm \frac{1}{1+ln x}, \frac{1}{1+lny}, \frac{1}{1+\ln z}

are

Solution:

Concept:

x

y

z

are in GP, then their natural logarithms are in AP, and adding a constant preserves AP, so reciprocals become HP.

Explanation:

Given

x, y, z > 1

are in GP.

For a GP,

y^2 = xz

Take natural logs:

2\ln y = \ln x + \ln z

Hence

\ln x, \ln y, \ln z

are in AP.

Add

1

to each term:

(1 + \ln x), (1 + \ln y), (1 + \ln z)

are also in AP.

Therefore, their reciprocals

\frac{1}{1 + \ln x}, \frac{1}{1 + \ln y}, \frac{1}{1 + \ln z}

are in HP.

Answer:

Option C: in HP.

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