Question

$\sum _{r=0}^m$${}^{n+r}{C}_n=$

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is A

$\sum _{r=0}^m{}^{n+r}{C}_n={}^n{C}_n+{}^{n+1}{C}_n+{}^{n+2}{C}_n+......+{}^{n+m-1}{C}_n+{}^{n+m}{C}_n$
We know,
${}^n{C}_n={}^{n+1}{C}_{n+1}=1$
Replace ${}^n{C}_n$ by ${}^{n+1}{C}_{n+1}$ for simplification purpose so that we can use the generalized formula
${}^n{C}_r+{}^n{C}_{r-1}={}^{n+1}{C}_r$
$\sum _{r=0}^m{}^{n+r}{C}_n={}^n{C}_n+{}^{n+1}{C}_n+{}^{n+2}{C}_n+.......{}^{n+m-1}{C}_n+{}^{n+m}{C}_n$
=${}^{n+1}{C}_{n+1}+{}^{n+1}{C}_n+{}^{n+2}{C}_n+.......{}^{n+m-1}{C}_n+{}^{n+m}{C}_n$
=${}^{n+2}{C}_{n+1}+{}^{n+2}{C}_n+.......{}^{n+m-1}{C}_n+{}^{n+m}{C}_n$
=${}^{n+3}{C}_{n+1}+.......{}^{n+m-1}{C}_n+{}^{n+m}{C}_n$
=${}^{n+m}{C}_{n+1}+{}^{n+m}{C}_n$
=${}^{n+m+1}{C}_{n+1}$