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  4. If x is a positive real number different from 1, then the number (1/1+√x), (1/1-x), (1/1-√x), ldots are in

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Q. If $x$ is a positive real number different from $1,$ then the number $\frac{1}{1+\sqrt{x}}, \frac{1}{1-x}, \frac{1}{1-\sqrt{x}}, \ldots$ are in

Sequences and Series

Solution:

Since $\frac{1}{1-x}-\frac{1}{1+\sqrt{x}}=\frac{1-(1-\sqrt{x})}{1-x}=\frac{\sqrt{x}}{1-x}$
And $\frac{1}{1-\sqrt{x}}-\frac{1}{1-x}=\frac{1+\sqrt{x}-1}{1-x}=\frac{\sqrt{x}}{1-x}$
Therefore, given numbers are in $A.P$.