Question

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

• A. A.P.
• B. G.P.
• C. H.P.
• D. No definite sequence

Answer 1

Answer: (A)

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 $A.P$.