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Question: Coefficient of \(x^{n}\) in the expansion of \((1 - 2x)^{- 1/2}\)...

Coefficient of xnx^{n} in the expansion of (1−2x)−1/2(1 - 2x)^{- 1/2}

A

(2r)!(r!)2\frac{(2r)!}{(r!)^{2}}

B

(2r)!2r.(r!)2\frac{(2r)!}{2^{r}.(r!)^{2}}

C

(2r)!(r!)2.22r\frac{(2r)!}{(r!)^{2}.2^{2r}}

D

(2r)!2r.(r+1)!(r−1)!\frac{(2r)!}{2^{r}.(r + 1)!(r - 1)!}

Answer

(2r)!2r.(r!)2\frac{(2r)!}{2^{r}.(r!)^{2}}

Explanation

Solution

Coefficient of

xr=(−12)(−12−1)(−12−2)....(−12−r+1)r!(−2)rx^{r} = \frac{\left( - \frac{1}{2} \right)\left( - \frac{1}{2} - 1 \right)\left( - \frac{1}{2} - 2 \right)....\left( - \frac{1}{2} - r + 1 \right)}{r!}( - 2)^{r}

=1.3.5...(2r−1).(−1)r.(−1)r.2r2rr!=1.3.5...(2r−1)r!=(2r)!r!r!2r= \frac{1.3.5...(2r - 1).( - 1)^{r}.( - 1)^{r}.2^{r}}{2^{r}r!} = \frac{1.3.5...(2r - 1)}{r!} = \frac{(2r)!}{r!r!2^{r}}