Binomial Theorem for Positive Integral Index: Statement, Structure, Application-Based Results

If n , i.e., n is a positive integer, and $\mathrm{a},\mathrm{b}$ are real numbers, then the binomial theorem states:
$(a+b{)}^n=\left[{}^n{C}_r\times a^{(n-r)}\times b^{\prime }\right]$  (from $r=0$ to $n$ )

Understanding Binomial Coefficients
Each term in the expansion has a binomial coefficient ${}^n{C}_r$ , calculated as:

${}^n{C}_r=n!/(r!\times (n-r)!)$

These coefficients have interesting properties:

1. ${}^n{C}_r={}^n{C}_{(n-r)}$ (Symmetry)

2. ${}^n{C}_0={}^n{C}_n=1$

3. ${}^n{C}_1={}^n{C}_{(n-1)}=n$

The binomial expansion of $(a+b{)}^n$  has exactly $(n+1)$  terms.
Each term looks like: $T_{(r+1)}={}^n{C}_r\ast a^{(n-r)}\cdot b^r$

Examples

$(x+y{)}^3=x^3+3x^{2}y+3x{y}^2+y^3$
Important Properties

1. Sum of Coefficients:
$(a+b{)}^n$ if $(a=1,b=1)=(1+1{)}^n=2^n$

2. Alternating Sum of Coefficients:
$(a+b{)}^n$ at $a=1,b=-1(1-1{)}^n=0$

3. Middle Terms:

• If n is even: one middle term at $\mathrm{r}=\frac{n}{2}$
• If n is odd: two middle terms at $\mathrm{r}=\frac{(n-1)}{2}$ and $\frac{(n+1)}{2}$

1. Term Independent of $x$ in ${\left(x+\frac{1}{x}\right)}^n$

$T_{(r+1)}={}^n{C}_r\cdot x^{(n-2r)};\text{ set }n-2r=0\mathrm{}r=n/2$

2. Finding a Specific Term:
e.g., 5 th term in $(2x-3{)}^6$ :

${\mathrm{T}}_5={}^6{\mathrm{C}}_4\cdot (2\mathrm{x}{)}^2\cdot (-3{)}^4=4860{\mathrm{x}}^2$

 

Important Tips for JEE

1. Memorize the general term: $T_{(r+1)}={}^n{C}_r{}^{\ast }{a}^{(n-r)}\cdot b^r$

2. Practice term finding problems

3. Use symmetry of coefficients

4. Master coefficient tricks with substitutions

5. Avoid full expansion unless necessary