Important Results of Binomial Theorem for any Index - Practice Questions & MCQ

Important Results of Binomial Theorem for any  index

If the given series is

$
(1+\mathrm{x})^{\mathrm{n}}=1+\mathrm{nx}+\frac{\mathrm{n}(\mathrm{n}-1)}{2!} \mathrm{x}^2+\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2)}{3!} \mathrm{x}^3+\ldots \ldots+\frac{\mathrm{n}(\mathrm{n}-1)(\mathrm{n}-2) \ldots \ldots(\mathrm{n}-\mathrm{r}+1)}{\mathrm{r}!} \mathrm{x}^{\mathrm{r}} \ldots . .
$
In the above expansion replace ' $n$ ' with ' $-n$ '

$
\begin{aligned}
&(1+\mathrm{x})^{-\mathrm{n}}=1+(-\mathrm{n}) \mathrm{x}+\frac{(-\mathrm{n})((-\mathrm{n})-1)}{2!} \mathrm{x}^2+\frac{(-\mathrm{n})((-\mathrm{n})-1)((-\mathrm{n})-2)}{3!} \mathrm{x}^3+\ldots \ldots \\
& \ldots+\frac{(-\mathrm{n})((-\mathrm{n})-1)((-\mathrm{n})-2) \ldots . \cdot((-\mathrm{n})-\mathrm{r}+1)}{\mathrm{r}!} \mathrm{x}^{\mathrm{r}} \ldots \ldots \ldots \infty \\
& \Rightarrow(1+\mathrm{x})^{-\mathrm{n}}= 1-\mathrm{nx}+\frac{\mathrm{n}(\mathrm{n}+1)}{2!} \mathrm{x}^2-\frac{\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2)}{3!} \mathrm{x}^3+\ldots \ldots \\
& \ldots+(-1)^{\mathrm{r}} \frac{\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2) \ldots \ldots(\mathrm{n}+\mathrm{r}-1)}{\mathrm{r}!} \mathrm{x}^{\mathrm{r}} \ldots \ldots \ldots \infty
\end{aligned}
$
If -n is a negative integer (so that n is a positive integer), then we can re-write this expression as

$
=1-{ }^n C_1 x+{ }^{n+1} C_2 x^2-{ }^{n+2} C_3 x^3+\cdots+{ }^{n+r-1} C_r(-x)^r+\cdots
$

Now replace ' $x$ ' with ' $-x$ ' and ' $n$ ' with ' $-n$ ' in the binomial expansion $(1+x)^n$.

$
\begin{aligned}
& (1-x)^{-n}=1+n x+\frac{n(n+1)}{2!} x^2+\frac{n(n+1)(n+2)}{3!} x^3+\cdots \\
& +\frac{n(n+1)(n+2) \cdots(n+r-1)}{r!} x^r+\cdots
\end{aligned}
$
If -n is a negative integer (so that n is a positive integer), then we can re-write this expression as

$
=1+{ }^n C_1 x+{ }^{n+1} C_2 x^2+{ }^{n+2} C_3 x^3+\cdots+{ }^{n+r-1} C_r(x)^r+\cdots
$
Important Note:
The coefficient of $x^r$ in $(1-x)^{-n}$, (when $n$ is a natural number) is ${ }^{n+r-1} C_r$

Some Important Binomial Expansion
1. $(1+x)^{-1}=1-x+x^2-x^3+\cdots$
2. $(1-x)^{-1}=1+x+x^2+x^3+\cdots$
3. $(1+x)^{-2}=1-2 x+3 x^2-4 x^3+\cdots$
4. $(1-x)^{-2}=1+2 x+3 x^2+4 x^3+\cdots$