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…
Statement-1: $\sum_{r=0}^n(r+1)^n C_r=(n+2) 2^{n-1}$
Statement -2: $\quad \sum_{r=0}^n(r+1)^n C_r x^r=(1+x)^n+n x(1+x)^{n-1}$.
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Statement $-1$ is false, Statement $-2$ is true
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Statement $-1$ is true, Statement $-2$ is true, Statement $-2$ is a correct explanation for Statement $-1$
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Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is not a correct explanation for Statement $-1$.
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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 $-1$
Asked in: JEE Main 2008
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