A) \[AP\]
B) HP
C) \[GP\]
D) None of these
Correct Answer: B
Solution :
Let the common ratio of the\[GP\]be\[r\]. Then,\[y=xr\]and\[z=x{{r}^{2}}\]. \[\Rightarrow \] \[\ln y=\ln x+\ln r\]and\[\ln z=\ln x+2\ln \,\,r\] Putting\[A=1+\ln x,\,\,D=\ln r\] Then, \[\frac{1}{1+\ln x}=\frac{1}{A}\] \[\frac{1}{1+\ln y}=\frac{1}{1+\ln xr}=\frac{1}{1+\ln x+\ln r}=\frac{1}{A+D}\] and\[\frac{1}{1+\ln z}=\frac{1}{1+\ln x+2\ln r}=\frac{1}{A+2D}\] Here we see that\[A,\,\,A+D\]and\[A+2D\]are in\[AP\]. \[\therefore \]\[\frac{1}{A},\,\,\frac{1}{A+D}\]and\[\frac{1}{A+2D}\]are in HP. Therefore, \[\frac{1}{1+\ln x},\,\frac{1}{1=\ln y},\,\frac{1}{1+\ln \,z},\] are in HP.
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