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  4. The sum displaystyle∑r=0n(r+1) Cr2 is equal to :

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Q. The sum $\displaystyle\sum_{r=0}^n(r+1) C_r^2$ is equal to :

Binomial Theorem

Solution:

$\displaystyle \sum_{r=0}^n(r+1) C_r^2$
$ (1+x)^n=C_0+C_1 x+\ldots \ldots \ldots . .+C_n x^n$
Multiply by $x$ & then differentiate
$ (1+x)^n+x . n(1+x)^{n-1}=C_0+2 C_1 x+\ldots \ldots \ldots \ldots(n+1) C_n x^n ....$(i)
$ (x+1)^n=C_0 x^n+C_1 x^{n-1}+\ldots \ldots \ldots \ldots+C_n \ldots \ldots \ldots $ (ii)
Multiply (i) & (ii) & equate the coefficient of $x ^n$ on both side
$C_0{ }^2+2 C_1{ }^2+\ldots \ldots . .+(n+1) C_n{ }^2={ }^{2 n} C_n+n \cdot{ }^{2 n-1} C_{n-1}=2 \cdot{ }^{2 n-1} C_{n-1}+n \cdot{ }^{2 n-1} C_{n-1}$
$=\frac{(n+2)(2 n-1) !}{n !(n-1) !}$