Mathematics GENERAL AND MIDDLE TERM IN BINOMIAL EXPANSION
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GENERAL AND MIDDLE TERM IN BINOMIAL EXPANSION

Important Terms In Binomial :

`(A) text(General Term :)`

`(T_(r+1))^(th)` term is called as general term in `(x + y)^n` and general term is given by

`T_(r+1)=text()^n C_r.x^(n-r) . y^r`


`(B) text(Term Independent of x :)`

It means term containing `x^0` .


`(C)text( Middle Term :)`

Let `T_m` is middle term is expansion `(x + y)^n` then

`text(Case I : lf n is odd, then number of terms will be even so there is two middle terms)`
`quadquadquadquadquad((n+1)/2)^(th)` and `((n+3)/2)^(th)`.

`text(Case II : lf n is even, then number of terms will be odd so only one term is middle term)`
`quadquadquadquadquad(n/2+1)^(th)`.


`text(Note :)`

1. Binomial coefficient of middle term is greatest.

2. If indices of a and b are positive integers.
Then, free from radical terms =Terms which are integers
:. Number of non-integral terms = Total terms- Number of integral terms

3. If indices of a and b both are not positive integers.
Then, free from radical terms =Rational terms - Integral terms

4. Number of irrational terms = Total terms- Number of rational terms.

Numerically Greatest Term in `(x+y)^n` :

`T_(r+1)` term is said to be numerically greatest for a given value of `x, y` provided `T_(r+1) ge T_r` and

`T_(r+1) ge T_(r+2)=> (T_(r+1))/(T_r) ge 1` as well as `(T_(r+1))/(T_(r+2)) ge 1`

`(T_(r+1))/T_r=(text()^nC_r(x)^(n-r)y^r)/(text()^nC_(r-1)(x)^(n-r+1)y^(r-1))=((n-r+1))/r. underbrace|(y/x)|_(text(Numerically greatest term
is required))`

`text(Shortcut Method : )`

To find the greatest term (numerically) in the expansion of `(x + y)^n`.

`(x+y)^n = y^n(1 + x/y)^n`

`r= (|x/y|(n+1))/(|x/y| +1)`

Case I : If me Integer, then `T_r` and `T_(r + 1)` are the greatest terms and both are equal (numerically).

Case II : If `m ∉ ` Integer, then `T_([m] + 1)` is the greatest term, where `[·]` denotes the greatest integer function.

`text(Greatest Coefficient : )`

(i) If n is even, then greatest coefficient is `text()^nC_(n/2)`

(ii) If n is odd, then greatest coefficients are `text()^nC_((n -1)/2)` and `text()^nC_((n + 1)/ 2) .`
 
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