Q54 / 105 · JEE Main 2008 · Mathematics

Statement-1: $\sum_{r=0}^n(r+1)^n C_r=(n+2) 2^{n-1}$ Statement -2: $\quad \sum_{r=0}^n(r+1)^n C_r x^r=(1+x)^n+n x(1+x)^{n-1}$.

  • A
    Statement $-1$ is false, Statement $-2$ is true
  • Correct answer
    B
    Statement $-1$ 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.

Solution

$$ \begin{aligned} & \sum_{r=0}^n(r+1){ }^n C_r=\sum_{r=0}^n r{ }^n C_r+{ }^n C_r \\ & =\sum_{r=0}^n r \frac{n}{r}{ }^{n-1} C_{r-1}+\sum_{r=0}^n{ }^n C_r=n 2^{n-1}+2^n \\ & =2^{n-1}(n+2) \end{aligned} $$ Statement $-1$ is true $$ \begin{aligned} & \sum(r+1)^n C_r x^r=\sum r{ }^n C_r x^r+\sum{ }^n C_r x^r \\ & =n \sum_{r=0}^n{ }^{n-1} C_{r-1} x^r+\sum_{r=0}^n{ }^n C_r x^r=n x(1+x)^{n-1}+(1+x)^n \end{aligned} $$ Substituting $x=1$ $$ \sum(r+1)^n C_r=n 2^{n-1}+2^n $$ Hence Statement $-2$ is also true and is a correct explanation of Statement $-1$.

Hence, the correct answer is option: 'B'.

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