First slide

Binomial theorem for positive integral Index

Question

If $C_{0}, C_{1}, C_{2}, C_{3}, \dots, C_{n}$ are the binomial coe11 ictenis mt e expans of $\frac{C_{0}}{1} + \frac{C_{2}}{3} + \frac{C_{4}}{5} + \frac{C_{6}}{7} + \dots$ has equal to

Moderate

A

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

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

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

none of these

Solution

From illustration 14 and 15, we have
and $\begin{aligned} \frac{C_{0}}{1} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + \dots + \frac{C_{n}}{n + 1} = \frac{2^{n + 1} - 1}{n + 1} \\ \frac{C_{0}}{1} - \frac{C_{1}}{2} + \frac{C_{2}}{3} - \dots + (- 1)^{n} \frac{C_{n}}{n + 1} = \frac{1}{n + 1} \end{aligned}$
Adding these two, we get
$\begin{aligned} 2 (\frac{C_{0}}{1} + \frac{C_{2}}{3} + \frac{C_{4}}{5} + \dots \dots .) = \frac{2^{n + 1}}{n + 1} \\ \frac{C_{0}}{1} + \frac{C_{2}}{3} + \frac{C_{4}}{5} + \dots = \frac{2^{n}}{n + 1} \end{aligned}$

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