Binomial Theorem
and Principle of
Mathematical Induction
An algebric expression consisting of two terms with positive and
negative sign between them is called binomial expression.
Binomial Theorem for Positive Integer
If n is any positive integer, then
( x + a )n = n C0x n + n C1x n − 1a + n C2x n − 2a 2 + ...+ n Cn a n .
n
i.e. ( x + a )n = ∑ n
Cr x n − r a r …(i)
r=0
where, x and a are real numbers and n C0 , n C1 , n C2 , K , n Cn are called
binomial coefficients.
Also, here Eq. (i) is called Binomial theorem.
n!
n
Cr = for 0 ≤ r ≤ n.
r !( n − r )!
Properties of Binomial Theorem for Positive Integer
(i) Total number of terms in the expansion of ( x + a )n is ( n + 1) i.e.
finite number of terms.
(ii) The sum of the indices of x and a in each term is n.
(iii) The above expansion is also true when x and a are complex
numbers.
(iv) The coefficient of terms equidistant from the beginning and the
end are equal. These coefficients are known as the binomial
coefficients i.e. n Cr = n Cn − r , r = 0, 1, 2, ... , n.
(v) The values of the binomial coefficients steadily increase to
maximum and then steadily decrease.
(vi) In the binomial expansion of ( x + a )n , the r th term from the end
is ( n − r + 2)th term from the beginning.
, (vii) If n is a positive integer, then number of terms in ( x + y + z )n is
( n + 1)( n + 2)
.
2
Some Special Cases
(i) ( x − a )n = n C0 x n − n C1x n −1a + n C2x n − 2a 2 − n C3 x n −3 a3
+ ... + ( − 1)n n Cn a n
n
∑ ( −1) Cr ⋅ x n − r ⋅ a r
r n
i.e. ( x − a )n =
r=0
(ii) (1 + x )n = n C0 + n C1x + n C2x 2 + ... + n Cr x r + K + n Cn x n
n
i.e. (1 + x )n = ∑ n Cr ⋅ x r
r=0
(iii) (1 − x ) = C0 − C1x + n C2x 2 − n C3 x3 + ... + ( −1)r n Cr x r
n n n
+ ... + ( −1)n n Cn x n
n
i.e. (1 − x )n = ∑ ( −1)r n
Cr ⋅ x r
r=0
(iv) The coefficient of x r in the expansion of (1 + x )n is n Cr and in the
expansion of (1 − x )n is ( −1)r n Cr .
(v) (a) ( x + a )n + ( x − a )n = 2 ( n C0x n a 0 + n C2x n − 2a 2 + K )
(b) ( x + a )n − ( x − a )n = 2 ( n C1x n −1a + n C3 x n − 3 a3 + K )
(vi) (a) If n is odd, then ( x + a )n + ( x − a )n and ( x + a )n − ( x − a )n
n + 1
both have the same number of terms equal to .
2
n
(b) If n is even, then ( x + a )n + ( x − a )n has + 1 terms.
2
n
and ( x + a ) − ( x − a ) has terms.
n n
2
General Term in a Binomial Expansion
(i) General term in the expansion of (x + a )n is
Tr +1 = n Cr x n − r a r
(ii) General term in the expansion of ( x − a )n is
Tr +1 = ( −1)r n Cr x n − r a r
and Principle of
Mathematical Induction
An algebric expression consisting of two terms with positive and
negative sign between them is called binomial expression.
Binomial Theorem for Positive Integer
If n is any positive integer, then
( x + a )n = n C0x n + n C1x n − 1a + n C2x n − 2a 2 + ...+ n Cn a n .
n
i.e. ( x + a )n = ∑ n
Cr x n − r a r …(i)
r=0
where, x and a are real numbers and n C0 , n C1 , n C2 , K , n Cn are called
binomial coefficients.
Also, here Eq. (i) is called Binomial theorem.
n!
n
Cr = for 0 ≤ r ≤ n.
r !( n − r )!
Properties of Binomial Theorem for Positive Integer
(i) Total number of terms in the expansion of ( x + a )n is ( n + 1) i.e.
finite number of terms.
(ii) The sum of the indices of x and a in each term is n.
(iii) The above expansion is also true when x and a are complex
numbers.
(iv) The coefficient of terms equidistant from the beginning and the
end are equal. These coefficients are known as the binomial
coefficients i.e. n Cr = n Cn − r , r = 0, 1, 2, ... , n.
(v) The values of the binomial coefficients steadily increase to
maximum and then steadily decrease.
(vi) In the binomial expansion of ( x + a )n , the r th term from the end
is ( n − r + 2)th term from the beginning.
, (vii) If n is a positive integer, then number of terms in ( x + y + z )n is
( n + 1)( n + 2)
.
2
Some Special Cases
(i) ( x − a )n = n C0 x n − n C1x n −1a + n C2x n − 2a 2 − n C3 x n −3 a3
+ ... + ( − 1)n n Cn a n
n
∑ ( −1) Cr ⋅ x n − r ⋅ a r
r n
i.e. ( x − a )n =
r=0
(ii) (1 + x )n = n C0 + n C1x + n C2x 2 + ... + n Cr x r + K + n Cn x n
n
i.e. (1 + x )n = ∑ n Cr ⋅ x r
r=0
(iii) (1 − x ) = C0 − C1x + n C2x 2 − n C3 x3 + ... + ( −1)r n Cr x r
n n n
+ ... + ( −1)n n Cn x n
n
i.e. (1 − x )n = ∑ ( −1)r n
Cr ⋅ x r
r=0
(iv) The coefficient of x r in the expansion of (1 + x )n is n Cr and in the
expansion of (1 − x )n is ( −1)r n Cr .
(v) (a) ( x + a )n + ( x − a )n = 2 ( n C0x n a 0 + n C2x n − 2a 2 + K )
(b) ( x + a )n − ( x − a )n = 2 ( n C1x n −1a + n C3 x n − 3 a3 + K )
(vi) (a) If n is odd, then ( x + a )n + ( x − a )n and ( x + a )n − ( x − a )n
n + 1
both have the same number of terms equal to .
2
n
(b) If n is even, then ( x + a )n + ( x − a )n has + 1 terms.
2
n
and ( x + a ) − ( x − a ) has terms.
n n
2
General Term in a Binomial Expansion
(i) General term in the expansion of (x + a )n is
Tr +1 = n Cr x n − r a r
(ii) General term in the expansion of ( x − a )n is
Tr +1 = ( −1)r n Cr x n − r a r
