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

Important Results of Binomial Theorem for any  index

If the given series is

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

$\begin{array}{rl}& (1+\mathrm{x}{)}^{-\mathrm{n}}=1+(-\mathrm{n})\mathrm{x}+\frac{(-\mathrm{n})((-\mathrm{n})-1)}{2!}{\mathrm{x}}^2+\frac{(-\mathrm{n})((-\mathrm{n})-1)((-\mathrm{n})-2)}{3!}{\mathrm{x}}^3+\dots \dots \\ & \dots +\frac{(-\mathrm{n})((-\mathrm{n})-1)((-\mathrm{n})-2)\dots .\cdot ((-\mathrm{n})-\mathrm{r}+1)}{\mathrm{r}!}{\mathrm{x}}^{\mathrm{r}}\dots \dots \dots \mathrm{\infty }\\ & \Rightarrow (1+\mathrm{x}{)}^{-\mathrm{n}}=1-\mathrm{nx}+\frac{\mathrm{n}(\mathrm{n}+1)}{2!}{\mathrm{x}}^2-\frac{\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2)}{3!}{\mathrm{x}}^3+\dots \dots \\ & \dots +(-1{)}^{\mathrm{r}}\frac{\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2)\dots \dots (\mathrm{n}+\mathrm{r}-1)}{\mathrm{r}!}{\mathrm{x}}^{\mathrm{r}}\dots \dots \dots \mathrm{\infty }\end{array}$
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 $(1+x{)}^n$.

$\begin{array}{rl}& (1-x{)}^{-n}=1+nx+\frac{n(n+1)}{2!}{x}^2+\frac{n(n+1)(n+2)}{3!}{x}^3+\cdots \\ & +\frac{n(n+1)(n+2)\cdots (n+r-1)}{r!}{x}^r+\cdots \end{array}$
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. $(1+x{)}^{-2}=1-2x+3x^2-4x^3+\cdots$
4. $(1-x{)}^{-2}=1+2x+3x^2+4x^3+\cdots$