Question

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

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

Answer 1

The correct option is B Statemet -1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

We have
${\sum }_{r=0}^n(r+1){}^n{C}_r{x}^r={\sum }_{r=0}^{n}r{}^n{C}_r{x}^r+{\sum }_{r=0}^n{}^n{C}_r{x}^r={\sum }_{r=1}^{n}r.{\frac{n}{r}}^{n-1}{C}_{r-1}{x}^r+(1+x{)}^n=nx{\sum }_{r=1}^n{}^{n-1}{C}_{r-1}{x}^{r-1}+(1+x{)}^n=nx(1+x{)}^{n-1}+(1+x{)}^n=$RHS
$\therefore$ Statement 2 is correct.
Putting x=1, we get
${\sum }_{r=0}^n(r+1{)}^n{c}_r=n.2^{n-1}+2^n=(n+2){.2}^{n-1}$
$\therefore$ Statement 1 is also true and statement 2 is a correct explanation for statement 1.