General Term
If $n$ is any natural number and a,b are any numbers, then
$\displaystyle \small (a+b)^{n}=^{n}C_{0}a^{n}+^{n}C_{1}a^{n-1}b+^{n}C_{2}a^{n-2}b^{2}$$\displaystyle \small +...+^{n}C_{r}a^{n-r}b^{r}+...+^{n}C_{n}b^{n}$
In the expansion of $(a+b)^{n}$, we find that the first term is $^{n}C_{0}a^{n}$, second term is $^{n}C_{1}a^{n-1}b$ and so on.
On looking at the pattern of successive terms, we find that $(r+1)^{th}$ term is $^{n}C_{r}a^{n-r}b^{r}$.
This term is also called the general term of the expansion $(a+b)^{n}$. It is denoted by $T_{r+1}$.
Thus, $T_{r+1}=^{n}C_{r}a^{n-r}b^{r}$
Special cases
(i) In the expansion of $(a-b)^{n}$, $T_{r+1}=((-1)^{r})^{n}C_{r}a^{n-r}b^{r}$
(ii) In the expansion of $(1+x)^{n}$, $T_{r+1}=^{n}C_{r}x^{r}$
(iii) In the expansion of $(1-x)^{n}$, $T_{r+1}=((-1)^{r})^{n}C_{r}x^{r}$
Middle Terms
Since the binomial expansion of $(a-b)^{n}$ contains $(n+1)$ terms,
(i) if $n$ is even, then number of terms in the expansion is $(n+1)$, which is odd. Therefore, there is only one middle term $\displaystyle \small \left ( \frac{n+1+1}{2} \right )^{th}$ i.e. $\displaystyle \small \left ( \frac{n}{2}+1 \right )^{th}$ term.
(ii) if $n$ is odd, then number of terms in the expansion is $(n+1)$, which is even. Therefore, there are two middle terms $\displaystyle \small \left ( \frac{n+1}{2} \right )^{th}$ term and $\displaystyle \small \left ( \frac{n+1}{2}+1 \right )^{th}$ term.
Constant Term or Term Independent
In the expansion of $\displaystyle \small \left ( x+\frac{1}{x} \right )^{2n}$, the middle term is $\displaystyle \small \left ( \frac{2n+1+1}{2} \right )^{th}$ i.e. $\displaystyle \small (n+1)^{th}$ term as $2n$ is even.
It is given by $\displaystyle \small ^{2n}C_{n}x^{n}\left ( \frac{1}{x} \right )^{n}=^{2n}C_{n}$ (constant)
This term is called the term independent of x or the constant term.
If $n$ is any natural number and a,b are any numbers, then
$\displaystyle \small (a+b)^{n}=^{n}C_{0}a^{n}+^{n}C_{1}a^{n-1}b+^{n}C_{2}a^{n-2}b^{2}$$\displaystyle \small +...+^{n}C_{r}a^{n-r}b^{r}+...+^{n}C_{n}b^{n}$
In the expansion of $(a+b)^{n}$, we find that the first term is $^{n}C_{0}a^{n}$, second term is $^{n}C_{1}a^{n-1}b$ and so on.
On looking at the pattern of successive terms, we find that $(r+1)^{th}$ term is $^{n}C_{r}a^{n-r}b^{r}$.
This term is also called the general term of the expansion $(a+b)^{n}$. It is denoted by $T_{r+1}$.
Thus, $T_{r+1}=^{n}C_{r}a^{n-r}b^{r}$
Special cases
(i) In the expansion of $(a-b)^{n}$, $T_{r+1}=((-1)^{r})^{n}C_{r}a^{n-r}b^{r}$
(ii) In the expansion of $(1+x)^{n}$, $T_{r+1}=^{n}C_{r}x^{r}$
(iii) In the expansion of $(1-x)^{n}$, $T_{r+1}=((-1)^{r})^{n}C_{r}x^{r}$
Middle Terms
Since the binomial expansion of $(a-b)^{n}$ contains $(n+1)$ terms,
(i) if $n$ is even, then number of terms in the expansion is $(n+1)$, which is odd. Therefore, there is only one middle term $\displaystyle \small \left ( \frac{n+1+1}{2} \right )^{th}$ i.e. $\displaystyle \small \left ( \frac{n}{2}+1 \right )^{th}$ term.
(ii) if $n$ is odd, then number of terms in the expansion is $(n+1)$, which is even. Therefore, there are two middle terms $\displaystyle \small \left ( \frac{n+1}{2} \right )^{th}$ term and $\displaystyle \small \left ( \frac{n+1}{2}+1 \right )^{th}$ term.
Constant Term or Term Independent
In the expansion of $\displaystyle \small \left ( x+\frac{1}{x} \right )^{2n}$, the middle term is $\displaystyle \small \left ( \frac{2n+1+1}{2} \right )^{th}$ i.e. $\displaystyle \small (n+1)^{th}$ term as $2n$ is even.
It is given by $\displaystyle \small ^{2n}C_{n}x^{n}\left ( \frac{1}{x} \right )^{n}=^{2n}C_{n}$ (constant)
This term is called the term independent of x or the constant term.
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