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

Answer: (A)

$\sum _{r=0}^n(r+1)C_r^2$
$(1+x{)}^n=C_0+C_{1}x+\dots \dots \dots ..+C_n{x}^n$
Multiply by $x$ & then differentiate
$(1+x{)}^n+x.n(1+x{)}^{n-1}=C_0+2C_{1}x+\dots \dots \dots \dots (n+1)C_n{x}^n....$(i)
$(x+1{)}^n=C_0{x}^n+C_1{x}^{n-1}+\dots \dots \dots \dots +C_n\dots \dots \dots$ (ii)
Multiply (i) & (ii) & equate the coefficient of $x^n$ on both side
$C_0{}^2+2C_1{}^2+\dots \dots ..+(n+1)C_n{}^2={}^{2n}{C}_n+n\cdot {}^{2n-1}{C}_{n-1}=2\cdot {}^{2n-1}{C}_{n-1}+n\cdot {}^{2n-1}{C}_{n-1}$
$=\frac{(n+2)(2n-1)!}{n!(n-1)!}$