UPES B.Tech Admissions 2025
ApplyRanked #42 among Engineering colleges in India by NIRF | Highest Package 1.3 CR , 100% Placements | Last Date to Apply: 30th July | Limited seats available in select program
Important Results of Binomial Theorem for any Index is considered one the most difficult concept.
52 Questions around this concept.
If the expansion in powers of of the function
:
If 0 < < 1, then the first negative term in the expansion of
is:
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. $(1-x)^{-2}=1+2 x+3 x^2+4 x^3+\cdots$
"Stay in the loop. Receive exam news, study resources, and expert advice!"