If `n` is any positive integer, then `( x+a)^n = text()^n C_0 x^n + text()^n C_1 x^(n-1) a+.....+ text()^n C_n a^n = sum_(r=0)^n text()^n C_r x^(n-r) a^r` where `x` and `a` are real (complex) numbers.

(i) In the expansion of `(x +a)^n`, it contains `(n + 1)` terms.

(ii) In the expansion of `(x +a)^n`, the sum of the powers of `x` and `a` in each term is equal to `n`.

(iii) The coefficient of terms equidistant from the beginning and the end are equal.

(iv) The values of the binomial coefficients steadily increase to maximum and then steadily decrease.

(v) `(x-a)^n = text()^n C_0 x^n - text()^n C_1 x^(n-1)a+.....+ (-1)^n \ \ text()^n C_n a^n`

(vi) `(1+x)^n = text()^n C_0 + text()^n C_1 x + text()^n C_2 x^2+............+ text()^n C_n x^n`

From the Pascal's triangle, it is clear that each coefficient of any row is obtained by adding two coefficients of preceding row, one on the immediate left and other on the immediate right and each row has `1` on both extreme sides.

Let `(r + 1)`th term be the general term in the expansion of `(x + a )^n`

Then `T_(r+1) = text()^(n)C_r x^(n-r) a^r`



If expansion is `(x- a)^n` then the general term is `T_(r+1) = (-1)^n* text()^(n)C_r x^(n-r) a^r`

The middle term in the expansion of `(a + x)^n` is as follows:

`"Case I :"` lf `n` is even, then `(n/2 +1 )` th term is the middle term.

`"Case II :"` If `n` is odd then `( n+1 )/2`th term and `(n+3)/2` th term are the middle terms.

In the binomial expansion of `(x +a )^n`, rth term from the end is `(bar ( n + 1) - r + 1)` i.e. `(n -r +2)` th term from the beginning.

Write down general term in the expansion of `(x + a)^n`.

i.e. . `T_( r+1) = text()^(n)C_r x^(n-r) a^r` .. (i)

For making a term independent of x, we put `r = n` in Eq. (i).

So, we get `text()^(n)C_n a^n` which is independent of x.

For making a term independent of a, we put r = 0 in Eq. (i).

So, we get `text()^(n)C_0 x^n` , which is independent of a.

`"NOTE-"` An independent term is also known as constant term

If `T_r` and `T_(r + 1)` are the rth and `(r + 1)`th terms in the
expansion of `(1 + x )^n` , then

`(T_(r+1))/(T_r) = ( text()^(n)C_(r) x^r )/(text()^(n)C_(r-1) x^(r-1) ) = (n-r +1)/r x`

Let numerically, `T_(r + 1)` be the greatest term in the above
expansion. Then,

`T_(r+1) ge T_r ` or `(T_(r+1) )/(T_r) ge 1`

`:. (n-r +1)/r |x| ge 1` or ` r le ( n+1)/( (1+ |x| ) ) |x| ` .............(i)


Now, substiuting the values of n and x in Eq. (i), we get
`r le m + f` or `r le m`, where m is a positive integer and f is a
fraction- such that `0 < f < 1`.

When n is even, then `T_(m+1)` is the greatest term and when n is.
odd, then `T_m` and `T_(m+1)` are the greatest terms and both are
equal.

(i) lf `n` is even, `text()^(n)C_(r)` is greatest when `r = n/2` i.e. greatest coefficient is ` text()^(n)C_(n/2)`.

(ii) If `n` is odd , `text()^(n)C_(r)` is greatest when `r = ( n-1 )/2` or `r = (n+1)/2`

i.e. greatest coefficients ` text()^(n) C_((n-1)/2)` and ` text()^(n) C_((n+1)/2)` .

In the binomial expansion of `( 1 + x )^n`,

`(1 + x)^n = text()^(n)C_(0) + text()^(n)C_(1) x + text()^(n)C_(2) x^2 + ... + text()^(n)C_(r) x^r + ... text()^(n)C_(n) x^n` , where

`text()^(n)C_(0) , text()^(n)C_(1) ,......text()^(n)C_(n)` are the coefficients of various powers of x,
are called binomial coefficient and it is also written as

`C_0 , C_1 ,....................,C_n` or `( ( n), (0) ) + ( ( n ), (1) ) + ..... + ( (n), (n) )`


(i) ` text()^(n)C_(r) = text()^(n)C_(s) => r = s` or `r+s = n => text()^(n)C_(r) + text()^(n)C_(r-1) = text()^(n-1)C_(r)`

(ii) `text()^(n+1)C_(r+1) = (n+1)/(r+1) text()^(n)C_(r)`

(iii) `(text()^(n)C_(r))/( text()^(n)C_(r-1)) = (n-r +1 )/r`

(iv) `C_0 + C_1 +C_2 + ......+ C_n = 2^n`

(v) `C_0 - C_2 +C_4 + ...... = C_1 + C_3 +C_5 + .........= 2^(n-1)`

(vi) ` C_0 - C_1 +C_2 -C_3 + .....+ (-1)^n * C_n = 0`

(vii) `C_(0)^2 + C_(1)^2 + C_(2)^2 + ......+ C_(n)^2 = text()^(2n)C_(n) = ( (2n)! )/( (n!)^2 )`

(vii) `C_0 C_r + C_1 C_(r+1) + ....+ C_(n-r) C_n = text()^(2n)C_(n-r) = ( (2n)! )/( (n-r)! (n+r)! ) `

(ix) `C_1 -2 C_2 +3 C_3 - .........= 0`

(x) `C_0 + 2C_1 +3 C_2 + .......+ (n+1)* C_n = (n+2) 2^(n-1)`

(xi ) `C_0 + C_1/2 x + C_2/3 x^2 + C_3/4 x^3 + ........+ (C_n)/(n+1) * x^n = ( (1+x)^(n+1) -1 )/( (n+1) n ) `


(xii ) `C_0 -C_2 + C_4 -C_6 +.........= (sqrt 2)^n cos \ (n pi )/4`

(xii) `C_1 -C_3 + C_5 -C_7 + .......= (sqrt 2)^n sin \ (n pi )/4`

(i) lf n is even, ` text()^(n)C_(r)` is greatest when `r = n/2` i.e., greatest

coefficient is `text()^(n)C_(n/2)`.

(ii) If n is odd, `text()^(n)C_(r)` is greatest when `r = (n-1)/2` or `r = ( n+1)/2`

i.e, greatest coeffiecient are ` text()^(n)C_((n-1)/2)` and `text()^(n)C_((n+1)/2)`.

In the binomial expansion of `( 1 + x )^n`,

`(1 + x)^n = text()^(n)C_(0) + text()^(n)C_(1) x + text()^(n)C_(2) x^r + .. text()^(n)C_(n) x^n` where

`text()^(n)C_(0) , text()^(n)C_(1) ............., text()^(n)C_(n)`
are the coefficientnts of various powers of x,
arc called binomial coefficient and it is also written as

`C_ 0 , C_1 , ............. C_n` or `( ( n), (0) ) + ( (n), (1) ) + ........+ ( (n), (n) )`

(i) `text()^(n)C_(r) = text()^(n)C_(s) => r = s ` or `r +s = n => text()^(n)C_(r) + text()^(n)C_(r-1) = text()^(n+1)C_(r)`

(ii) `text()^(n+1)C_(r+1) = (n+1)/(r+1) text()^(n)C_(r)`

(iii) `( text()^(n)C_(r))/( text()^(n)C_(r-1)) = (n-r +1)/r`

(iv) `C_0 + C_1 +C_2 +............+ C_n = 2^n`

(v) `C_0 + C_2 + C_4 +....= C_1 +C_3 + C_5 + ......= 2^(n-1)`

(vi) `C_0 -C_1 +C_2 - C_3 + .......+ (-1)^n * C_n = 0`

(vii) `C_(0)^2 + C_(1)^2 + C_(2)^2 +....+ C_(n)^2 = text()^(2n)C_(n) = ( (2n)! )/( (n!)^2 )`

(viii) `C_0 C_r +C_1 C_(r+1) + ........+ C_(n-r) C_n = text()^(2n)C_(n-r) = ( (2n)! )/( ( n-r)! (n+r)! ) `

(ix) `C_1 -2C_2 + 3C_3 - .....= 0`

(x) `C_0 + 2C_1 +3 C_2 + .......+ (n+1) * C_n = (n+2) 2^(n-1)`

(xi ) `C_0 + C_1/2 x + C_2/3 x^2 + C_3/4 x^3 + .... + C_n/(n+1) * x^n = ( (1+x)^(n+1) -1 )/( (n+1) n) `

(xii) `C_0 - C_2 +C_4 - C_6 + ........ = (sqrt 2)^n cos (n pi)/4`

(xiii ) `C_1 -C_3 +C_5 -C_7 + ..... (sqrt 2)^n sin (n pi)/4`

`"(i) R-f Factor Relation"`

Here, we are going to discuss problems involving

`(sqrt A +B )^n = l + f`

where, l and n are positive integers.

`0 le f le 1 , | A-B^2 | = k` and ` | sqrt A -B | < 1`

`"(ii) Divisibility Problem"`

In the expansion of `( 1 + α )^n = 1 + text()^(n)C_(1) α + text()^(n)C_(2) α^2 + ... + text()^(n)C_(n) α^n`,

we can conclude that `( 1 + α )^(n) -1 = text()^(n)C_(1) α + text()^(n)C_(2) α^2 + ....+ text()^(n)C_(n) α^n`

is divisible by ` α` i.e. it is a multiple of `α `

Let n be a rational number and x be a real number such that ` | x | < 1` , then

`( 1+x )^n = 1+ nx + ( n (n-1) )/(2!) - x^2 + ( n (n-1) (n-2) )/(3!) x^3 + ...`

e.g. The expansion of `(x+a)^n = a^n (1+x/a)^n`

`= a^n { 1+ n (x/a) + (n (n-1) )/(2!) (x/a)^2 + ........ }`

The above expansion is valid only, when ` | x/a | < 1` .

This expansion is infinite. It terminates after `(n +1)` terms, if `n` is `+ ve` integer, otherwise it goes to infinity.

(i) `(1+x)^n = text()^(n)C_(0) x^0 + text()^(n)C_(1) x + text()^(n)C_(2) x^2 + .....+ text()^(n)C_(r) x^r + ....+ text()^(n)C_(n-1) x^(n-1) + text()^(n)C_(n) x^n`

`= sum_(r=0)^n text()^(n)C_(r) x^r`

(ii) ` (1-x)^n = text()^(n)C_(0) x^0 - text()^(n)C_(1) x + text()^(n)C_(2) x^2 - ....+ text()^(n)C_(n-1) (-1)^(n-1) x^(n-1) + text()^(n)C_(n) (-1)^n x^n`

`= sum_(r=0)^n (-1)^r * text()^(n)C_(r) x^r`

(iii) ` (x+a)^n = text()^(n)C_(0) x^n + text()^(n)C_(1) x^(n-1) a + text()^(n)C_(2) x^(n-2) a^2

+ .....+ text()^(n)C_(r) x^(n-r) a^r + .........+ text()^(n)C_(n) x^0 a^n`

(iv) `(a+x)^n = text()^(n)C_(0) a^n x^0 + text()^(n)C_(1) a^(n-1) x + text()^(n)C_(2) a^(n-2 ) x^2

+ .......+ text()^(n)C_(r) a^(n-r) * x^r + ..... + text()^(n)C_(n) x^n`

`= sum_(r=0)^n text()^(n)C_(r) a^(n-r) * x^r`

(v) `(a-x)^n = text()^(n)C_(0) a^n x^0 - text()^(n)C_(1 ) a^(n-1) x + text()^(n)C_(2) a^(n-2) x^2`

`- .... + (-1)^r .text()^(n)C_(r) text()^(n)C_(r) a^(n-r) x^r + .....+ (-1)^n .text()^(n)C_(n) * a^(n-r) x^n`

`= sum_(r=0)^n (-1)^r * text()^(n)C_(r) a^(n-r) x^r`

(vi) ` (a + x)^n + (a-x)^n = 2 ( text()^(n)C_(0) a^n x^0 + text()^(n)C_(2) a^(n-2) x^2 + .... ) `

`=2` (Sum of the terms at odd places)

(vii) ` (a+x)^n - (a-x)^n = 2 ( text()^(n)C_(1) a^(n-1) x + text()^(n)C_(3) a^(n-3) x^3 + .... ) `

`=2` (Sum of the terms at even places)

(viii) `(1+x)^(-1) = 1 -x + x^2 - x^3 .....+ (-1)^r x^r + ...`

(ix) `(1-x)^(-1) = 1 +x + x^2 + x^3 .....+ x^r + ....`

(x) `( 1+x)^(-2) = 1 -2x + 3x^2 - 4 x^3 .....+ (-1)^r (r+1) x^r + ....`

(xi) ` (1-x)^(-2) = 1 + 2x + 3x^2 + 4x^3 .....+ (r+1) x^r +.....`

(xii) ` ( 1+x)^(-3) = 1 -3x + 6x^2 -....+ (-1)^r (r+1) (r+2) x^r + ....`

(xiii) `(1-x)^(-3) =1 + 3x + 6x^2 + ....+ (r+1) (r+2) x^r + ....`

(xiv) If the coefficients of rth, `(r + 1)`th, `(r + 2)`th terms of

`(1+x)^n` are i n AP , then `n^2 - (4r +1) n+ 4r^2 = 2`