Question

Statement-l: $\sum _{r=0}^n(r+1{)}^n{C}_r=(n+2)2^{n-1}$
Statement-2: $\sum _{r=0}^n(r+1{)}^n{C}_r{x}^r=(1+x{)}^n+nx(1+x{)}^{n-1}$

• A. Statement-1 is true, Statement-2 is true; Statement -2 is not a correct explanation for Statement-1.
• B. Statement-1 is true, Statement-2 is false.
• C. Statement-1 is false, Statement-2 is true.
• D. Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

Answer 1

The correct option is D Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

$\sum _{r=0}^n(r+1{)}^n{C}_r{x}^r=\sum _{r=0}^{n}r.C_r{x}^r+\sum _{r=0}^n{C}_r.x=nx\sum _{r=1}^n{C}_{r-1}x+\sum _{r=0}^n{C}_{r}x=nx(1+x{)}^{n-1}+(1+x{)}^n$ ... (i)
$\Rightarrow$ Statement-2 is true.
Putting $x=1$ in (i), we get $\sum _{r=0}^n(r+1)n_{C_r=}(n+2){.2}^{n-1}\Rightarrow$ Statement-l is also true and followed by I.