Binomial Theorem

The Expansion of (a + b)n
If   $n$   is any positive integer, then

$(a + b)^n = a^n + {_nC_1}a^{n - 1}b + {_nC_2}a^{n - 2}b^2 + \, \cdots \, + {_nC_r}a^{n - r}b^r + \, \cdots \, + b^n$
 

Where
${_nC_r}$ = combination of n objects taken r at a time.
 

Some Example of Binomial Expansion
$(a + b)^2 = a^2 + 2ab + b^2$

$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

$(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$

$(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5$

$(a + b)^6 = a^6 + 6a^5b + 15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6$

$(a + b)^7 = a^7 + 7a^6b + 21a^5b^2 + 35a^4b^3 + 35a^3b^4 + 21a^2b^5 + 7ab^6 + b^7$
 

The coefficient of terms can also be found by
 

$C = \dfrac{\text{coefficient of previous term } \, \times \, \text{ exponent of } \, a \, \text{ of previous term}}{\text{exponent of } \, b \, \text{ of previous term } + \, 1}$

 

Properties of Binomial Expansion

  1. The first term and last term of the expansion are   $a^n$   and   $b^n$,   respectively.
  2. There are   $n + 1$   terms in the expansion.
  3. The sum of the exponents of   $a$   and   $b$   in any term is   $n$.
  4. The exponent of   $a$   decreases by   $1$   from   $n$   to   $0$.
  5. The exponent of   $b$   increases by   $1$   from   $0$   to   $n$.
  6. The coefficient of the second term and the second from the last term is   $n$.

Pascal's Triangle
Pascal's triangle can be used to find the coefficient of binomial expansion.

(a + b)0   :   1
(a + b)1   :   1   1
(a + b)2   :   1   2   1
(a + b)3   :   1   3   3     1
(a + b)4   :   1   4   6     4     1
(a + b)5   :   1   5   10   10   5     1
(a + b)6   :   1   6   15   20   15   6     1
(a + b)7   :   1   7   21   35   35   21   7   1
 

rth term of (a + b)n

$r^{th} \, \text{ term } = \dfrac{n!}{(n - r + 1)!(r - 1)!}a^{n - r + 1} \, b^{r - 1}$

or

$r^{th} \, \text{term} = {_nC_m} a^{n - m} b^m$

where m = r - 1
 

For n = even, the middle term is at

$r = \frac{1}{2}n + 1$

 

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