Question

The value of $\sum _{r=0}^n$ ${}^{n+r}{C}_r$ is equal to

• A. ${}^{2n+1}{C}_n$
• B. ${}^{2n}{C}_{n-1}$
• C. ${}^{2n}{C}_{n+1}$
• D. ${}^{2n+1}{C}_{n-1}$

Answer 1

The correct option is A ${}^{2n+1}{C}_n$

The given series can be written as
${}^n{C}_0{+}^{n+1}{C}_1{+}^{n+2}{C}_2+.....{+}^{2n}{C}_n=\left\{{}^{n+1}{C}_0{+}^{n+1}{C}_1\right\}{+}^{n+2}{C}_2+.....{+}^{2n}{C}_n=\left\{{}^{n+2}{C}_1{+}^{n+2}{C}_2\right\}+.....{+}^{2n}{C}_n=\left\{{}^{n+3}{C}_2{+}^{n+3}{C}_3\right\}{+}^{n+4}{C}_4+....{+}^{2n}{C}_n{=}^{n+4}{C}_3{+}^{n+4}{C}_4+....{+}^{2n}{C}_n$
Proceeding the same way finally we get the sum as
${}^{2n+1}{C}_n$