First slide

Binomial theorem for positive integral Index

Question

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

Easy

A

$\frac{1}{(n+ 1) x}$
B

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

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

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

Solution

$\begin{array}{l} S = 1 + \frac{n}{2!} x + \frac{n (n - 1)}{3!} x^{2} + \dots + \frac{x^{n}}{n + 1} \\ x \cdot S = x + 1 + \frac{n}{2!} x^{2} + \frac{n (n - 1)}{3!} x^{3} + \dots + \frac{x^{n + 1}}{n + 1} \\ (n + 1) x (S) = (n + 1) x + \frac{(n + 1) n}{2!} x^{2} + \frac{(n + 1) n (n - 1)}{3!} x^{3} + \dots + \frac{(n + 1) x^{n + 1}}{n + 1} \\ 1^{n^{n + 1}} C_{1} x +^{n + 1} C_{2} x^{2} + \dots^{n + 1} C_{n + 1} x^{n + 1} - 1 \\ = (1 + x)^{n + 1} - 1 \\ S = \frac{(1 + x)^{n + 1} - 1}{x (n + 1)} \end{array}$

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