Question

The sum$\sum _{r=0}^n(r+1)C_r^2$ is equal to :

• A. $\frac{(n+2)(2n-1)!}{n!(n-1)!}$
• B. $\frac{(n+2)(2n+1)!}{n!(n-1)!}$
• C. $\frac{(n+2)(2n+1)!}{n!(n+1)!}$
• D. $\frac{(n+2)(2n-1)!}{n!(n+1)!}$

Answer 1

The correct option is A $\frac{(n+2)(2n-1)!}{n!(n-1)!}$

$a{C}_0^2+(a+d)C_1^2+(a+2d)C_2^2...(a+nd)C_n^2=\left(\frac{2a+nd}{2}\right){}^{2n}{\text{C}}_n$

Using the formula mentioned

Here $a=1$ and $d=1$

$\sum _{r=0}^n(r+1)C_2^r=\left(\frac{2\times 1+n\times 1}{2}\right)\times {}^{2n}{C}_n$

$=\left(\frac{n+2}{2}\right)\times \frac{2n!}{n!\times n!}$

$=\frac{n+2}{2}\times 2n\times \frac{(2n-1)!}{n\times (n-1)!\times n!}$

$=\frac{(n+2)\times (2n-1)!}{n!\times (n-1)!}$