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  4. If n ≥ 2 is a positive integer, then the sum of the series n+1 C2+2( 2 C2+ 3 C2+ 4 C2+ ldots .+ n C2) is:

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Q. If $n \geq 2$ is a positive integer, then the sum of the series ${ }^{n+1} C_{2}+2\left({ }^{2} C_{2}+{ }^{3} C_{2}+{ }^{4} C_{2}+\ldots .+{ }^{n} C_{2}\right)$ is:

JEE MainJEE Main 2021Binomial Theorem

Solution:

${ }^{n+1} C_{2}+2\left({ }^{2} C_{2}+{ }^{3} C_{2}+{ }^{4} C_{2}+\ldots \ldots .+{ }^{n} C_{2}\right)$
${ }^{n+1} C_{2}+2\left({ }^{3} C_{3}+{ }^{3} C_{2}+{ }^{4} C_{2}+\ldots \ldots .+{ }^{n} C_{2}\right)$
$\left\{\right.$ use $\left.{ }^{n} C_{r+1}+{ }^{n} C_{r}={ }^{n+1} C_{r}\right\}$
$={ }^{n+1} C_{2}+2\left({ }^{4} C_{3}+{ }^{4} C_{2}+{ }^{5} C_{3}+\ldots \ldots+{ }^{n} C_{2}\right)$
${ }^{n+1} C_{2}+2\left({ }^{5} C_{3}+{ }^{5} C_{2}+\ldots \ldots .+{ }^{n} C_{2}\right) \vdots$
$={ }^{n+1} C_{2}+2\left({ }^{n} C_{3}+{ }^{n} C_{2}\right)$
$={ }^{n+1} C_{2}+2 \cdot{ }^{n+1} C_{3}$
$=\frac{(n+1) n}{2}+2 \cdot \frac{(n+1)(n)(n-1)}{2.3}$
$ =\frac{n(n+1)(2 n+1)}{6}$