A) \[{{\left( \frac{1+x}{1-x} \right)}^{n}}\]
B) \[{{\left( \frac{2x}{1+x} \right)}^{n}}\]
C) \[{{\left( \frac{1+x}{2x} \right)}^{n}}\]
D) \[{{\left( \frac{1-x}{1+x} \right)}^{n}}\]
Correct Answer: A
Solution :
\[{{(1+x)}^{n}}={{\,}^{n}}{{C}_{0}}+{{\,}^{n}}{{C}_{1}}x+{{\,}^{n}}{{C}_{2}}{{x}^{2}}+....+{{\,}^{n}}{{C}_{n}}{{x}^{n}}+....\] If x is replaced by \[-\left( \frac{2x}{1+x} \right)\]and n is -n. Hence \[{{\left[ 1-\left( \frac{2x}{1+x} \right) \right]}^{-n}}=1+(-n)\left[ -\left( \frac{2x}{1+x} \right) \right]\]\[+\frac{(-n)(-n-1)}{2!}{{\left( -\frac{2x}{1+x} \right)}^{2}}+....\] \[{{\left[ \frac{1-x}{1+x} \right]}^{-n}}=1+n\left( \frac{2x}{1+x} \right)+\frac{n(n+1)}{2}{{\left( \frac{2x}{1+x} \right)}^{2}}+....\] or \[{{\left[ \frac{1+x}{1-x} \right]}^{n}}=1+n\text{ }\left( \frac{2x}{1+x} \right)+\frac{n(n+1)}{2}{{\left( \frac{2x}{1+x} \right)}^{2}}+....\]
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