First slide

Binomial theorem for positive integral Index

Question

$C_{0} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + .. + \frac{C_{n}}{n+1} =$

Moderate

A

$\frac{2^{n+1} -1}{n+1}$
B

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

$\frac{2^{n}}{n+1}$
D

$\frac{1}{n+1}$

Solution

$\begin{array}{l} C_{0} + \frac{C_{1}}{2} x + \frac{C_{2}}{3} x^{2} + \dots + \frac{C_{n}}{n + 1} x^{n} = \frac{(1 + x)^{n + 1} - 1}{(n + 1) x} \\ \Rightarrow C_{0} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + \dots + \frac{C_{n}}{n + 1} = \frac{2^{n + 1} - 1}{n + 1} . \end{array}$

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