Question

If $p$ and $q$ are the coeffcients of $x^n$ in $(1+x{)}^{2n}$ and $(1-4x{)}^{-1/2},|x|<\frac{1}{4}$ then

• A. $p=q$
• B. $p=2q$
• C. $q=2p$
• D. $p+q{=}^{2n}{C}_n$

Answer 1

The correct option is A $p=q$

The coefficient of $x^{n}in{(1+x)}^{2n}$ is $p{=}^{2n}{C}_n$.
The coefficient of $x^{n}in{(1-4x)}^{-1/2}$ is
$\Rightarrow q=4^n\times \frac{1\times 3\times 5\times ...\times (2n-1)}{2^n\times n!}$
$\Rightarrow q=\frac{1\times 2\times 3\times 4\times ...\times \left(2n\right)2^n}{2\times 4\times 6\times ...\times \left(2n\right)n!}$
$\Rightarrow q=\frac{\left(2n\right)!}{n!\times n!}{=}^{2n}{C}_n$
Hence, $p=q$.