First slide

Binomial theorem for positive integral Index

Question

$If (1 + x)^{n} = \sum_{r = 0}^{n} C_{r} x^{r}$ then $(1 + \frac{C_{1}}{C_{0}}) (1 + \frac{C_{2}}{C_{1}}) \dots (1 + \frac{C_{n}}{C_{n - 1}})$ is equal to

Moderate

A

$\frac{n^{n - 1}}{(n - 1)}$
B

$\frac{(n + 1)^{n - 1}}{(n - 1)!}$
C

$\frac{(n + 1)^{n}}{n!}$
D

$\frac{(n + 1)^{n + 1}}{n!}$

Solution

We have $(1 + \frac{C_{1}}{C_{0}}) (1 + \frac{C_{2}}{C_{1}}) \dots (1 + \frac{C_{n}}{C_{n - 1}})$
$\begin{array}{l} = (1 + \frac{n}{1}) [1 + \frac{\frac{n (n - 1)}{2!}}{n}] \dots (1 + \frac{1}{n}) \\ = \frac{(1 + n)}{1} \cdot \frac{(1 + n)}{2} \cdot \frac{(1 + n)}{3} \dots \frac{(1 + n)}{n} = \frac{(n + 1)^{n}}{n!} \end{array}$

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