Question

The value of expression
$C_0^2-C_1^2+C_2^2-......{(-1)}^n(n+1)C_n^2$ is

• A. $0$ if $n$ is odd
• B. $(n+1){(-1}^n$ if $n$ is odd
• C. ${(-1)}^{n/2}(\frac{n}{2}+1)\frac{n!}{\left(\frac{n}{2}\right)!\left(\frac{n}{2}\right)!}$
• D. none of these

Answer 1

Answer: (A), (C)

$0$ if $n$ is odd
${(-1)}^{n/2}(\frac{n}{2}+1)\frac{n!}{\left(\frac{n}{2}\right)!\left(\frac{n}{2}\right)!}$

When $n$ is odd, we take $n=2m-1$, so that
$S=C_0^2-C_1^2+C_2^2-......{(-1)}^{2m-1}\left(2m\right)C_{2m-1}^2$ ........(1)
Using ${{}^{2m-1}C}_r={{}^{2m-1}C}_{2m-1-r}$, we rewrite (1) as
$S=-2m{C}_0^2+(2m-1)C_1^2+....+C_{2m-1}^2$ .......(2)
Adding (1) and (2), we get
$2S=(1-2m)[C_0^2-C_1^2+C_2^2-......{(-1)}^{2m-1}{C}_{2m-1}^2]$
$(1-2m)\left(0\right)=0\Rightarrow S=0$
When $n$ is even, we take $n=2m$
$S=C_0^2-C_1^2+C_2^2-......{(-1)}^{2m-1}\left(2m\right)C_{2m-1}+{(-1)}^{2m}(2m+1)C_{2m}^2$ .......(3)
Using ${{}^{2m}C}_r={{}^{2m}C}_{2m-r}$, we rewrite (3) as
$S=(2m+1)C_0^2-\left(2m\right)C_1^2+(2m-1)C_2^2-.....+{(-1)}^{2m-1}{C}_{2m}^2$ ........(4)
Adding (3) and (4), we get
$2S=(2m+2)[C_0^2-C_1^2+C_2^2-......{(-1)}^{2m}{C}_{2m}^2]$
$\Rightarrow S=(m+1){(-1)}^m({{}^{2m}C}_{m)}$
$={(-1)}^{n/2}(\frac{n}{2}+1)\frac{n!}{\left(\frac{n}{2}\right)!\left(\frac{n}{2}\right)!}$