First slide

Binomial theorem for positive integral Index

Question

If $C_{r} =^{n} C_{r},$ then $C_{0} - \frac{1}{3} C_{1} + \frac{1}{5} C_{2} \dots upto (n + 1)$ term equal

Moderate

A

$\int_{0}^{1} x (1 - x)^{n} d x$
B

$\int_{0}^{1} {(1 - x^{2})}^{n} d x$
C

0
D

$1 + \int_{0}^{1} x (1 + x)^{n} d x$

Solution

$\begin{array}{l} C_{0} - \frac{1}{3} C_{1} + \frac{1}{5} C_{2} - \dots \\ = \int_{0}^{1} [C_{0} - C_{1} x^{2} + C_{2} x^{4} - \dots] d x = \int_{0}^{1} {(1 - x^{2})}^{n} d x \end{array}$

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