First slide

Binomial theorem for positive integral Index

Question

$If S_{n} = \sum_{i = 0}^{n} \frac{1}{^{n} C_{r}} and t_{n} = \sum_{r = 0}^{n} \frac{r}{^{n} C_{r}} then \frac{t_{n}}{S_{n}} =$

Moderate

A

$\frac{1}{2} n$
B

$\frac{2 n - 1}{2}$
C

$n - 1$
D

$\frac{1}{2} n - 1$

Solution

$\begin{array}{l} t_{n} = \sum_{r = 0}^{n} \frac{r}{^{n} C_{r}} \\ = \sum_{r = 0}^{n} \frac{n - r}{^{n} C_{n - r}} = \sum_{r = 0}^{n} \frac{n - r}{^{n} C_{r}} = n \sum_{r = 0}^{n} \frac{1}{^{n} C_{r}} - \sum_{r = 0}^{n} \frac{r}{^{n} C_{r}} \\ = {ns}_{n} - t_{n} \Rightarrow 2 t_{n} = {ns}_{n} \Rightarrow \frac{t_{n}}{S_{n}} = \frac{n}{2} \end{array}$

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