Binomial Theorem Formulas





BINOMIAL THEOREM, EXPONENTIAL AND LOGARITHMIC SERIES

 

Binomial theorem is used to describe algebraic expression of power of a binomial, in elementary algebra. In other words binomial theorem can also be called as a quick way of expanding a binomial expression that have been raised to certain power.

 

Some of the important formulas of binomial theorem are listed below :-

 

1. Binomial theorem for positive integral index:-

 

(x + y)^n = X^n + C (n, 1) X ^{n- 1}y + C (n, 2) X^{n- 2}y^2 + \cdots + C (n, r) X ^{n- r}y^r + \cdots + C (n, n-1)XY{n- 1} + C(n, n)y^n.

 

It can be represented as: (x + y)^n =\sum_{i = 0} ^n C (n, r) X^{n- r}y^r

 

Some particular cases:

 

2. Replacing ‘y’ by ‘-y’ , we have

 

(x- y)^n = C (n, o) X^n- C (n, 1) x^{n- 1}y + C (n, 2) X^{n- 2} y^2 \cdots + (-1)^r C(n, r) X^{n- r}y^r + \cdots + -(-1)^{n- 1}C(n, n- 1) xy^{n- 1} + (-1)^n C(n, n)X^\circ y^n

 

It can be represented as: (x- y)^n = \sum_{i = 0}^n (-1)^r C(n, r)x^{n- r}y^r

 

3. Replacing ‘x’ by ‘1’ and ‘y’ by ‘x’, we have

 

(1 + x)^n = C (n, o) + C (n, 1) x + C (n, 2) x^2 + \cdots + C(n, r) x^r + \cdots C(n, n- 1) x^{n- 1} + C (n, n) xn

 

Or, (1 + x)^n = \sum_{I = 0}^n C(n, r)x^r

 

 

Properties of Binomial Expansion (x + y)^n:-

 
a.    There are (n + 1) terms in the expansion.

b.    In each term, sum of the indices of ‘x’ and ‘y’ is equal to ‘n’.

c.    In any term, the lower suffix of ‘c’ is equal to the index of ‘y’, and the index of x = n – (lower suffix of c).

d.    Because C(n, r) = C(n, n-r) so we have : C(n, n) C(n, 1) = C(n, n-1)C(n, 2) = C(n, n-2) it follows that the coefficients of terms equidistant from the beginning and the ends are equal.

 
1.    General terms:

 
(r + 1)^n term from beginning is (x + y)^n is called general term, and it is denoted by t_{r + 1} = x^{n- r} y^r

 
2.    Middle terms:

 
It depends upon the value of ‘n’.

 

Case I : when ‘n’ is even, then total number of terms is (x + y)^n is odd. So there is only one middle term
i.e. t_{n/2+ 1} = C(n, n/2) x^{n- n/2}y^{n/2}

 
case II: when ‘n’ is odd, then total number of terms in (x + y)^n is even. So there are two middle terms i.e. (n + 1)/2 th and (n + 3)/2 the i.e. t_{\frac{n + 1}{2}} = C(n, (n- 1)/2)x^{n- (n-1)/2}.y^{(n- 1)/2} and t_{(n + 1)/2 + 1} = C(n, (n + 1)/n)x^{n- (n + 1)/2}.y^{(n + 1)/2}

 

 
3.    Some other relations

  • Expression of exponential

i.    e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots + \cdots to \infty
ii.    e^{-x} = 1- x + \dfrac{x^2}{2!}- \dfrac{x^3}{3!} + \dfrac{x^4}{4!}- \cdots + \cdots to \infty
iii.    e^1 = 1 + 1 + \dfrac{1}{2!} + \dfrac{1}{3!} + \cdots to \infty
iv.    e^{-1} = 1- 1 + \dfrac{1}{2!}- \dfrac{1}{3!} + \cdots to \infty
v.    a^x = 1 + x(\log a) + \dfrac{(x \log a)^2}{2!} + \cdots to \infty

 

  • Expansion of logarithmic function

i.    \log(1 + x) = x- \dfrac{x^2}{2} + \dfrac{x^3}{3} + \dfrac{x^4}{4} + \cdots to \infty
ii.    \log(1- x) = -x- \dfrac{x^2}{2}- \dfrac{x^3}{3}- \cdots to \infty
iii.    \log(1 + x) + \log(1- x) = 2\left(x + \frac{x^3}{3} + \frac{x^5}{5} + \frac{x^7}{7} + \cdots\right)

 

  • Binomial Coefficient

 
The coefficient C(n, 0), C(n, 1)……….C(n, n) in the expansion of (a + x)^n is known as binomial coefficient and are devoted by C_0, C_1, C_2, \cdots Cn logarithmic respectively.

 
i.    C_0 + C_1 \cdots + C_n = 2^n (sum of binomial coefficient)

 
ii.    C_0 + C_2 + C_4 + \cdots = C_1 + C_3 +C_5 + \cdots = 2^{n- 1}



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