Binomial theorem for any index

Binomial theorem for any index for entrance exams.

1. BINOMIAL THEOREM FOR ANY INDEX
Let n be a rational number and x be a real number such that |x| < 1 , then
(1+x)
n
= 1+ nx + x
2
+ . . . + x
r
+ ... terms upto ∞ ......(1)
Observations:
o General term of the series (1+x)
-n
= Tr+1 = (-1)
r


x
r

o General term of the series (1-x)
-n
= Tr+1 = x
r

o If first term is not 1, then make first term unity in the following way:
(a+ x)
n
= a
n
(1+)
n
if < ||< 1
IMPORTANT EXPANSIONS
· (1+ x)
-1
= 1- x +x
2
–x
3
+ . . . + (-1)
r
x
r
+. . .
· (1 - x)
-1
= 1+ x +x
2
+x
3
+ . . .+ x
r
+ . . .
· (1+ x)
-2
= 1- 2x +3x
2
–4x
3
+ . . .+ (-1)
r
(r+1)x
r
+. . .
· (1 - x)
-2
= 1+ 2x +3x
2
+4x
3
+ . . .+ (r+1)x
r
+. . .
· (1+x)
-3
= 1- 3x +6x
2
–10x
3
+. . .+ (-1)
r
x
r
+. . .
· (1-x)
-3
= 1+ 3x +6x
2
+10x
3
+ . . .+ x
r
+. . .

· In general coefficient of x
r
in (1 – x)
– n
is
n + r –1
Cr .
· (1 – x)
–p/q
= 1 + p/1! (x/q) + p(p+q)/2! (x/q)
2
+ ……. From (1)
· (1 + x)
–p/q
= 1 – p/1! (x/q) + p(p+q)/2! (x/q)
2
– …….
· (1 + x)
p/q
= 1 + p/1! (x/q) + p(p–q)/2! (x/q)
2
+ …….
· (1 – x)
p/q
= 1 – p/1! (x/q) + p(p–q)/2! (x/q)
2
– …….

Solved Example 1: If –1 < x < 1, show that (1 –x)
-2
= 1 + 2x + 3x
2
+ 4x
3
+
…..to ∞.
Solution: We know that if n is a negative integer or fraction
(1+x)
n
= 1 +
Provided –1 < x < 1

Putting n = –2 and –x in place of x, we get
(1+x)
2
= 1 +
= 1 + 2x + 3x
2
+ 4x
3
+ … to ∞.

Properties Of Binomial Expansion
1. There are (n + 1) terms in the expansion of (a + b)
n
, the first and the last
term being a
n
and b
n
respectively. If
n
Cx =
n
Cy, then either x = y or x + y = n

n
Cr =
n
Cn-r = n!/r!(n-r)! .(coeffts. Of terms equidistant from beginning and
the end are equal)
2. The general term in the expansion of (a + x)
n
is (r + 1)
th
term given as

2. Tr+1 =
n
Cra
n-r
x
r
.
Similarly the general term in the expansion of (x + a)
n
is given as
Tr+1 =
n
Crx
n-r
a
r
. The terms are considered from the beginning.
3. The binomial coefficient in the expansion of (a + x)
n
which are equidistant
from the
beginning and the end are equal i.e.
n
Cr =
n
Cn-r.
Note: Here we are using
n
Cr
+
n
Cr-1
=
n+1
Cr
this concept will be discussed later in this chapter.
Also, we have replace
m
C0
by
m+1
C0
because numerical value of both is same i.e. 1. Similarly we
replace
m
Cm
by
m+1
Cm+1
.
Properties of Binomial Coefficients
For the sake of convenience, the coefficients
n
Co,
n
C1, ...,
n
Cr, ...,
n
Cn are usually
denoted by C o, C1,..., Cr, ..., Cn respectively
Put x = 1 in (A) and get, 2
n
= Co + C1 + ... + Cn ... (D)
Also putting x = -1 in (A) we get,
0 = Co - C1 + C2 - C3 + ......
=> C0 + C2 + C4 + ...... + C1 + C3 + C5 +...... = 2
n
.
Hence Co + C2 + C4 +...... = C1 + C3 + C5 +...... = 2
n-1
.

n
Cr = .
n-1
Cr-1 = .
n-2
Cr-2 and so on.....
= , r = 1,2,3.....
EXAMPLE: If C0 , C1 , C2 ... Cn denote the binomial coeffts. In the expansion of
(1+x)
n
, prove that:
+ 2 + 3 +.........+n =
(ii) (C0+ C1)( C1+ C2)( C2+ C3)..........( Cn-1+ Cn) =
SOLUTION: use
= , r = 1,2,3.....
r. = n-r+1, r=1,2,3....n

3. = = n+(n-1)+.....+3+2+1
(ii) ( ).........() =
)(1+ ).........(1+) (either can take only L.H.S. &mult.
And divide by c0.c1.c2.....cn ) now use above result.
Illustration:
If (1 + x)
n
= C0 + C1x + C2x
2
+...+ Cnx
n
, then prove that C 0 + (Co + C1) + (C0 +
C1+ C2) + ... (C0 + C1 + C2 +...+ Cn-1) = n2
n-1
(where n is an even integer.
Solution:
C0 + (C0 + C1) + (C0 + C1 + C2) +... (C0 + C1 + C2 +...+ Cn-1)
= C0+(C0 + C1 + C2 +...+ Cn-1)+(C0 + C1)+(C0 + C1 + C2 +...+ Cn-2)+...
= (C0 + C1 + C2 +...+ Cn)+ (C0 + C1 + C2 +...+ Cn)+... n/2 times
= (n/2)2
n
= n . 2
n-1

Sum Of Binomial Coefficients

Putting x = 1 in the expansion (1+x)
n
=
n
C0 +
n
C1 x +
n
C2 x
2
+...+
n
Cx x
n
, we
get,
2
n
=
n
C0 +
n
C1 x +
n
C2 +...+
n
Cn.
We kept x = 1, and got the desired result i.e. ∑
n
r=0 Cr = 2
n
.
Note: This one is very simple illustration of how we put some value of x
and get the solution of the problem. It is very important how judiciously you
exploit this property of binomial expansion.
Illustration:
Find the value of C0 + C2 + C4 +... in the expansion of (1+x)
n
.
Solution:
We have,
(1 + x)
n
=
n
C0 +
n
C1 x +
n
C2 x
2
+...
n
Cn x
n
.
Now put x = -x; (1-x)
n
=
n
C0 -
n
C1 x +
n
C2 x
2
-...+ (-1)
n

n
Cn x
n
.

4. Now, adding both expansions, we get,
(1 + x)
n
+ (1-x)
n
= 2[
n
C0 +
n
C2 x
2
+
n
C4 x
4
+.........]
Put x = 1
=> (2
n
+0)/2 = C 0 + C2 + C4 +......
or C0 + C2 + C4 +......= 2
n-1


We have found the sum of binomial coefficients. But if these coefficients are
multiplied by some factors can we find the sum for such expressions?
Yes, we can often find it by creatively applying what we have learnt. Let
us see how differentiation and integration are useful in these situations.
Suppose we want to calculate the value of
∑
n
r=0 Cr i.e. 0.C0 + 1 C1 + 2C2 +...+ n
n
Cn
If we take a close look to the sum to be found, we find that coefficients
are multiplied with respective powers of x.
If we want to multiply the coefficient of x by its power differentiation is of
help. Hence differentiate both sides of
(1 + x)
n
=
n
C0 +
n
C1 x +
n
C2 x
2
+
n
C3 x
3
+...+
n
Cn x
n
, with respect to x we
get
n(1 + x)
n-1
= 1 C1 x
1-1
+ 2 C2 x
2-1
+...+ n Cn x
n-1

Put x = 1, we get, n 2
n-1
= + 1 C1 + 2 C2 +...+ n Cn.
Or, ∑
n
r=0 r Cr = n 2
n-1
, which is the answer.

Important: It has said earlier in this chapter that we should use and
exploit the property that x can take any value in the expansion of (1 + x)
n
.
Now, let us try to find the value of
C0 - C2 + C4-C6 +.........
Analysing the above, expression, we find that C 0, C2, C4 are all
coefficients of even powers of x.
Had it been like C0 + C2 + C4 +...... we could have evaluated simply by the
method described earlier or had it been like C 0-C1 + C2-C3 +... we could have
put x = -1 in the expansion of (1 + x)
n
and find the sum.

5. But here the case is different. The expression consists of coefficients of only
even powers. In such cases when there is alternative sign charge for the
coefficient, of same nature of powers (even or odd) use of I (iota) comes to
our rescue.
(1 + x)
n
= C0 + C1 x + C2 x
2
+......
(1 - x)
n
= C0 - C1 x + C2 x
2
+......
By adding (1 + x)
n
+ (1 - x)
n
= 2[C0 + C2 x
2
+ C4 x
4
+......].
Clearly use of √-1 (iota) will generate the given expression.
((1+i)
n
+ (1-i)
n
)/2 ] [C0 - C2 + C4 + ......]
1. Divisibility problems:
Let (1 + x)
n
= 1 +
n
C1x +
n
C2x
2
+...+
n
Cnx
n
.
In any divisibility problem, we have to identify x and n. The number by
which division is to be made can be, x, x
2
or x
3
, but the number in the base is
always expressed in form of (1+x).

Illustration:
Find the remainder when 7
103
is divided by 24.
Solution:
7
103
= 7(50 - 1)
51
= 7(50
51
-
51
C1 50
50
+
51
C2 50
49
- ... - 1)
= 7(50
51
-
51
C150
50
+...+
51
C5050) - 7 - 18 + 18
= 7(50
51
-
51
C150
50
+...+
51
C5050) - 25 + 18
=> remainder is 18.


Multinomial Expression
If such a case arises, then it is not called Binomial Expansion, it is called
Multinomial Expansion. If n ε N, then the general term of the multinomial
expansion (x
1
+ x
2
+ x
3
+...+ x
k
)
n
is (n!/a1!a2!...ak!) (x1
a1
,x2
a2
,..…xk
ak
), where
a1 + a2 + a3+......+ ak = n and a < a i < n, I = 1, 2, 3, ...... k and the total
number of terms in the expansion is
n+k-1
Cn-1.

6. Problem Related to Series of Binomial Coefficients in Which Each Term is a
Product of two Binomial Coefficients .
(a) If sum of lower suffices of binomial expansion in each term is the same
i.e.
n
C0
n
Cn +
n
C1
n
Cn-1 +
n
Cn-2 +...+
n
Cn
n
C0

i.e. 0 + n = 1 + (n-1) = 2 + (n-2) = n + 0.
Then the series represents the coefficients of xn in the multiplication of
the following two series
(1+x)
n
= C0 + C1x + C2x
2
+...+ Cnx
n

and (1+x)
n
= C0 + C1x + C2x
2
+...+ Cnx
n
.
Illustration:
Prove that C0Cr +C1Cr+1 +C2Cr+2 +...+Cn-r Cn =



(Similarily we can prove that C0C1 + C1C2 + C2C3 +...+ Cn-1 Cn =


Solution:
We have,
C0 + C1x + C2x
2
+ ... + Cnx
n
= (1+x)
n
... (1)
Also C0x
n
+ C2x
n-2
+...+ Cn = (x+1)
n
... (2)
Multiplying (1) and (2), we get
(C0 + C1x
2
+...+ Cnx
n
)(C0x
n
+ C1x
n-2
+ C2x
n-2
+...+ Cn) = (1+x)
2n
... (3)
Equating coefficient of x
n-r
from both sides of (3), we get
C0Cr + C1Cr+1 + C2Cr+2 +...+ Cn-rCn =
=

7. Illustration:
(i) Prove that Co
2
+ C1
2
+...+ Cn
2
= (2n)!/n!n! .
(ii) prove that Co
2
- C1
2
+...+(-1)
n
Cn
2
= if n is even
Solution:
(i) Since
(1 + x)
n
= C0 + C1x + C2x
2
+...+ Cnx
n
, ... (1)
(x + 1)
n
= C0x
n
+ C1x
n-2
+...+ Cn, ... (2)
(C0 + C1x + C2x
2
+...+ Cnx
n
)(C0x
n
+ C1x
n-1
+ C2x
n-2
+...+ Cn)=(1+x)
2n
.
Equating coefficient of x
n
, we get
C0
2
+ C1
2
+ C2
2
+...+ Cn
2
=
2n
Cn = (2n)!/n!n! .
(ii) (x+1)
n
(1-x)
n
= (C0x
n
+ C1x
(n-1)
+ C2x
n-2
+...+ Cn-1x+Cn )( C0 - C1x +
C2x
2
-C3x
3
+C4x
4
...+(-1)
n
Cnx
n
)=
(1-x
2
)
n
= C0 - C1x
2
+ C2x
4
-C3x
6
+...+(-1)
n
Cnx
2n

Equating the coeffts of x
n
on both sides
Co
2
- C1
2
+...+(-1)
n
Cn
2
= 0, if n is odd
If n is even , suppose (r+1)th term in the expansion of (1-x
2
)
n
contains x
n

, we have Tr+1 = (-1)
r
x
2r
it implies 2r =n i.e., r=n/2 hence proved.
Example: Find the coeffts. Of x
4
in the expansion of (1+x)
n
(1-x)
n
.
Deduce that C2 = C0C4 - C1C3 + C2C2 – C3 C1+ C4C0
SOLUTION: (1+x)
n
(1-x)
n
= (C0 + C1x + C2x
2
+C3x
3
+C4x
4
...+ Cnx
n
)( C0 - C1x +
C2x
2
-C3x
3
+C4x
4
...+(-1)
n
Cnx
n
)
Coeffts of x
4
on R.H.S. = C0C4-C1C3+C2C2-C3C1+C4C0
(1-x
2
)
n
= C0 - C1x
2
+ C2x
4
-C3x
6
+...+(-1)
n
Cnx
2n


Illustration:
If (1+x)
n
= , then prove that

m
Cr
n
C0 +
m
Cr-1
n
C1 +
m
Cr-2
n
C2 +...+
m
C1
n
Cr-1 +
m
C0
n
Cr =
m+n
Cr

8. where m, n, r are positive integers and r < m and r < n.

Solution:
(1+x)
n
=
n
C0 +
n
C1x +
n
C2x
2
+...+
n
Crx
r
+...+
n
Cnx
n
... (1)
and also
(1+x)
m
=
m
C0 +
m
C1x +
m
C2x
2
+...+
m
Crx
r
+...+
m
Cmx
m
... (2)
Multiplying (1) and (2), we get
(
n
C0 +
n
C1x +
n
C2x
2
+...+
n
Crx
r
+...+
n
Cnx
n
)x
(
m
C0 +
m
C1x +
m
C2x
2
+...+
m
Crx
r
+...+
m
Cmx
m
) = (1+x)
m+n

=
m+n
C0 +
m+2
C1x +
m+n
C2x
2
+...+
m+n
Crx
r
+...+
m+n
Cm+nx
m+n

Equating the coefficient of x
r
, we get

m
Cr
n
C0 +
m
Cr-1
n
C1 +
m
Cr-2
n
C2 +...+
m
C1
n
Cr-1 +
m
C0
n
Cr =
m+n
Cr

(b) If one series has constant lower suffices and other has varying lower
suffices
Illustration:
Prove that
n
C0.
2n
Cn -
n
C1
2n-2
Cn +
n
C2.
2n-4
Cn -...= 2
n
.
Solution:

n
C0.
2n
Cn -
n
C1
2n-2
Cn +
n
C2/
2n-4
Cn -...
= coefficient of x
n
in[
n
C0(1+x)
2n
-
n
C1(1+x)
2n-2
+
n
C2(1+x)
2n-4
- ...]
= coefficient of x
n
in
[
n
C0((1+x)
2
)
n
-
n
C1((1+x)
2
)
n-1
+
n
C2((1+x)
2
)
n-2
- ...]
= coefficient of x
n
in [(1+x)
2
- 1]
n

= coefficient of x
n
in (2x+x
2
)
n
= co-efficient of x
n
in x
n
(2+x)
n
= 2
n
.


Some Important Results

9. 1. Differentiating (1+x)
n
= C0 + C1x + C2x
2
+...+ Cnx
n
of both sides we
have,
n(1 + x)
n-1
= C1 + 2C2x + 3C3x
2
+...+
n
Cnx
n-1
. ... (E)
Put x = 1 in (E) so that n2
n+1
= C1 + 2C2 + 3C3 + ...+ nCn.
Put x = -1 in (E) so that 0 = C1 - 2C2 +...+ (-1)
n-1

n
Cn.
Differentiating (E) again and again we will have different results.
2. Integrating (1 + x)
n
, we have,
((1+x)
n+1
)/(n+1) + C = C 0x + C1x
2
/2 + C2x
3
/3 +.....+Cnx
n+1
/(n+1)
(where C is a constant)
For x = 0, we get C = -1/(n+1) .
Therefore ((1+x)
n+1
- 1)/(n+1) = C0x + C1x
2
/2 + C2x
3
/3
+.....+Cnx
n+1
/(n+1) ..... (F)
Put x = 1 in (F) and get
(2
n+1
- 1)/n+1= C 0 + C1/2 +...Cn/n+1 .
Put x = -1 in (F) and get, 1/n+1 = C0 - C1/2 + C2/3 - ......
Put x = 2 in (F) and get, (3
n+1
-1)/n+1 = 2 C 0 + 2
2
/2 C1 + 2
2
/3
C2 +...+ 2
n+1
/n+1 = Cn.
Problems Related to Series of Binomial Coefficients in Which Each Term
is a Product of an Integer and a Binomial Coefficient, i.e. In the Form
k.
n
Cr.

Illustration:
If (1+x)
n
= x
r
then prove that C 1 + 2C2 + 3C3 +...+ nCn = n2
n-1
.
Solution:
Method (i) : r
th
term of the given series
r
th
term of the given series, tr = nCr

tr = r × n/r ×
n-1
Cr-1 = n ×
n-1
Cr-1 (because
n
Cr =n/r .
n-1
Cr-1)
Sum of the series =

10. Put x = 1 in the expansion of (1 + x)
n-1
, so that
(
n-1
C0 +
n-1
C1 +...+
n-1
Cn-1) = 2n-1
=> = n.2n-1.
Method (ii) : By Calculus
We have (1 + x)
n
= C0 + C1x + C2x
2
+...+ Cnx
n
. ... (1)
Differentiating (1) w.r.t. x, we get
n(1 + x)
n-1
= C1 + 2C2x + 3C3 x
2
+...+ n Cnx
n-1
. ... (2)
Putting x = 1 in (2), we have, n 2
n-1
= C1 + 2C2 +....+ nCn. ... (3)

Illustration:
If (1+ x)
n
= x
r
then prove that C 0 + 2.C1 + 3.C2+...+(n+1)C n
=2
n-1
(n+2).
Solution:
Method (i): r
th
term of the given series
rth term of the given series
tr =
n
Cr-1 = [(r-1) + 1].
n
Cr-1

= (r-1)
n
Cr-1 +
n
Cr-1 = n.
n-1
Cr-2 +
n
Cr-1 (because
n
Cr-1 =n/(r-1) .
n-1
Cr-2)
Sum of the series =
= n[
n-1
C0 +
n-1
C1 +...+
n-1
Cn-1]+[
n
C0 +
n
C1 +...+
n
Cn] = n.2
n-1
+ 2n
= 2
n-1
(n+2).
Method (ii) by Calculus.
We have (1 + x)
n
= C0 + C1x + C2x
2
+...+ Cnx
n
. ... (1)
Multiplying (1) with x, we get
x(1+x)
n
= C0x + C1x
2
+ C2x
3
+...+ Cnx
n+1
. ... (2)
Differentiating (2) w.r.t. x, we have
(1 + x)
n
+ n(1 + x)
n-1
x = C0x + 2C1x
2
+...+ (n+1)C nx
n
...
(3)

11. Putting x = 1 in (3), we get
2
n
+ n.2
n-1
= C0 + 2C1 + 3C2 +...+ (n+1)Cn

=> C0 + 2C1 + 3C2 +...+ (n+1)Cn = 2
n-1
(n+2).

Problems Related to Series of Binomial Coefficient in Which Each Term is
Binomial Coefficient divided by an Integer, i.e. in the Form of
n
Cr/k.
Now, think how to find the following sum
C0/1 + C1/2 + C2/3 +......+ Cn/n+1 = ?
In this sum coefficients are divided by the respective power of x +
1. This expression can be achieved by Integrating the expansion of (1 +
x)
n
under proper limits.
= = = [ .....
{ use = . , and add & subtract }= [2
(n+1)
-1]
Similarily we can prove C0/1 - C1/2 - C2/3 +......+(-1)
n
Cn/n+1 =
1/(n+1)
L.H.S. = = =
[ ....
Or another method



Illustration:
If (1+x)
n
=
Solution:

12. Method (i) : r
th
term of the given series

Method (ii): By Calculus
(1+x)
n
= C0 + C1x + C2x
2
+...+ Cnx
n
... (1)
Integrating both the sides of (1) w.r.t. x between the limits 0 to x,
we get
... (2)
Substituting x = 1 in (2), we get 2
n+1
/n+1 = C0 + c1/2 + c2/3
+.....+ cn/n+1 .

Illustration:
If (1+x)n = x
r
,show that

Method I : r
th
term of the given series T r =

13. Method II : (By Calculus)
(1 + x)
n
= C0 + C1x + C2x
2
+...+ Cnx
n

=> x(1+x)
n
= C0x + C1x
2
+ C2x
3
+...+ Cnx
n+1
... (1)
Integrating both the sides of (1) with respect to x

Put x = 0, => k = 1/((n+1)(n+2))


Put x = 1,
.


Greatest Binomial Coefficient

To determine the greatest coefficient in the binomial expansion, (1+x)
n
,
when n is a positive integer. Coefficient of
(Tr+1/Tr) = Cr/Cr-1 = (n-r+1)/r = ((n+1)/r) - 1.
Now the (r+1)
th
binomial coefficient will be greater than the rth
binomial coefficient when, T r+1 > Tr

=> ((n+1)/r)-1 > 1 => (n+1)/2 >r. ....... (1)
But r must be an integer, and therefore when n is even, the greatest
binomial coefficient is given by the greatest value of r, consistent with
(1) i.e., r = n/2 and hence the greatest binomial coefficient is
n
Cn/2.
Similarly in n be odd, the greatest binomial coefficient is given when,
r = (n-1)/2 or (n+1)/2 and the coefficient itself will be
n
C(n+1)/2 or
n
C(n-
1)/2, both being are equal

14. Note: The greatest binomial coefficient is the binomial coefficient of
the middle term.

Illustration:
Show that the greatest the coefficient in the expansion of (x +
1/x)
2n
is (1.3.5...(2n-1).2
n
)/n! .
Solution:
Since middle term has the greatest coefficient.
So, greatest coefficient = coefficient of middle term
=
2n
Cn = (1.2.3...2
n
)/n!n! = (1.3.5...(2n-1).2
n
)/n!.

Numerically greatest term
To determine the numerically greatest term in the expansion of (a + x)
n
,
where n is a positive integer.
Consider

Thus


Note : {((n+1)/r) - 1} must be positive since n > r. Thus T r+1 will be
the greatest term if, r has the greatest value as per the equation (1).

Illustration:
Find the greatest term in the expansion of (3 -2x)
9
when x = 1.

15. Solution:
Tr+1/Tr = ((9-r+1)/r) . (2x/3) >1

i.e. 20 > 5r
If r = 4, then T r+1 = Tr and these are the greatest terms. Thus
4
th
and 5
th
terms are numerically equal and greater than any other term
and their value is equal 3
9
×
9
C3 × (2/3)
3
= 489888.

Illustration:
Find the greatest term in the expansion of (2 + 3x)
9
if x = 3/2.
Solution:
Here Tr+1/Tr = ((n-r+1)/r)(3x/2) = ((10 -r)/r)(3x/2), (where x =
3/2)
= ((10-r)/r)(3x/2)(3/2) = ((10 -r)/r).9/4 = (90-9r)/4r
Therefore Tr+1 > Tr, if 90 - 9r > 4r.
=> 90 > 13r => r < 90/13 and r being an integer, r = 6.
Hence Tr+1 = T7 = T6+1 =
9
C6 (2)
3
(3x)
6
= 3
13
.7/2.

Illustration:
Given that the 4
th
term in the expansion of (2 + (3/8)x)
10
has the
maximum numerical value, find the range of values of x for which this
will be true.
Solution:
Given 4
th
term in (2 + (3/8)x)
10
= 2
10
(1 + (3/16)x)
10
, is
numerically greatest
=>|T4/T3| > 1 and |T5/T4| < 1
=>
=>|x| > 2 and |x| < 64/21
=> x ε [-(64/21),-2]U[2,(64/21)].

16. Illustration:
Find the greatest term in the expansion of √3(1 + (1/√3))
20
.
Solution:
Let r
th
term be the greatest
Since Tr/Tr+1 = (r/(21-r)).√3.
Now Tr/Tr+1 > 1 => r > 21/(√3+1) ...... (1)
Now again T r/Tr+1 = (r/(21-r)) √3 < 1 => r < (22+√3)/(√3+1)
...... (2)
from (1) and (2)
22/(√3+1) < r < (22+√3)/(√3+1)
=> r = 8 is the greatest term and its value is √3 .
20
C7
(1/√3)
7
=
20
C7 (1/27).
Illustration:
Find the first negative term in the expansion of (1+x)
7/2
.
Solution: Tr+1 = this will be first negative term if
<0 it implies r>9/2 or r≥5 , hence 6
th
term is the first
negative term = -7/256 x
5
( by putting r=5).
Illustration:
If (1-x)
-n
= a 0+a1x+a2x
2
+a3x
3
+......., find the value of
a0+a1+a2+......+an.
Solution: use (1 -x)
-n
(1-x)
-
1
=(a0+a1x+a2x
2
+.....+anx
n
+...)(1+x+x
2
....+x
n
+....)
Coeffts. Of x
n
in (1-x)
-(n+1)
= a0+a1+a2+......+an
Tr+1=
=
Put r=n then coeffts. Of x
n
in (1-x)
-(n+1)
= a0+a1+a2+......+an
= = (2n)!/(n!)
2

17. Illustration:
Find the coefft. Of x
6
in (1+x+x2)
-3

Solution: (1+x+x2)
-3
={ }
-3
= =(1-x)
3
(1-x
3
)
-3

Coefft. Of x
6
is 6-3=3.

Particular Cases
We have (a + x)
n
= a
n
+
n
C1 a
n-1
x +
n
C2 a
n-2
x
r
+...+ x
n
. ...... (1)
(i) Putting x = -x in (1), we get
(a-x)
n
= a
n
-
n
C1 a
n-1
x +
n
C2 a
n-2
x
2
-
n
C2 a
n-3
x
3
+...+ (-1)
r

n
Cr a
n-
r
x
r
+...+ (-1)
n
x
n
.
(ii) Putting a = 1 in (1), we get,
(1+x)
n
=
n
C0 +
n
C1x +
n
C2x
2
+...+
n
Cr x
r
+...+
n
Cnx
n
. ... (A)
(iii) Putting a = 1, x = -x in (1), we get
(1-x)
n
=
n
C0-
n
C1 x +
n
C2 x
2
-
n
C3x
3
+... (-1)
r

n
Cr x
r
+... (-1)
n
nCnx
n
...
(B)

Tips to Remember
(a) Tr/Tr+1 = ((n-r+1)/r ).(x/a) for the binomial expansion of (a + x)
n
.
(b)
(n+1)
Cr =
n
Cr +
n
Cr-1.
(c) r
n
Cr = n
n-1
Cr-1

(d)
(e) When n is even,
(x + a)
n
+ (x - a)
n
= 2(x
n
+
n
C2 x
n-2
a
2
+
n
C4 x
n-4
a
4
+...+
n
Cn an).
When n is odd,
(x + a)
n
+ (x - a)
n
= 2(x
n
+
n
C2 x
n-2
a
2
+...+
n
Cn-1 x a
n-1
).
When n is even

18. (x + a)
n
- (x - a)
n
= 2(
n
C1 x
n-1
a +
n
C3 x
n-3
a
3
+...+
n
Cn-1 x a
n-1
).
When n is odd
(x + a)
n
- (x - a)
n
= 2(
n
C1 x
n-1
a +
n
C3 x
n-3
a
3
+...+
n
Cn a
n
).
Binomial Theorem(Approximation )

Illustration:
If p is nearly to q and n>1, show that = ,hence find
.
Solution: let p = q+h, where h is very small that its square & higher
powers may be neglected. Then , L.H.S.

= = =

-1

={ }={1+h/(nq)} [expanding
neglecting higher power of h]
R.H.S.
= = = 1+h/(nq) [ same reason]
Put p=99, q= 101 and n=6 in above result , we get =
599/601.

Illustration:
Assuming x to be so small that x
2
and higher power of x can be
neglected, show that:
is approx. Equal to 1- (305x/96).
Solution: =

= =(1-3x)(1-3x/32)(1-x/12) [by
using (1+x)
n
=1+nx] =(1-3x-3x/32)(1-x/12)=
(1-99x/32-x/12)=1-305x/96. [by neglecting higher power of x]
Similarily, we can find a and b , if = a+bx for all

19. small values of x , a= 1, b= -305/96 by equating first two terms
on the both sides.
SOME EXAMPLES
Example1 :
If a1, a2, a3 and a4 are the coefficients of any four consecutive
terms in the expansion of (1+x)
n
then prove that:
a1/(a1+a2) + a2/(a3+a4) = 2a2/(a2+a3)
Solution:
As a1, a2, a3 and a4 are coefficients of consecutive terms, then
Let a1 =
n
Cr
a2 =
n
Cr+1
a3 =
n
Cr+2 and
a4 =
n
Cr+3
Now a1/(a1+a2) =
n
Cr/(
n
Cr+
n
Cr+1) = 1/(1+((n-r)/(r+1))) =
(r+1)/(n+1)
Similarly, a2/(a2+a3) = (r+3)/(n+1)
Now a3/(a3+a4) + a1/(a1+a2) = (2r+4)/(n+1)
= 2(r+1)/(n+1) = 2a 2/(a2+a3) (Hence,
proved)
Example2 :
Find out which one is larger 99
50
+ 100
50
or 101
50
.
Solution:
Let's try to find out 101
50
- 99
50
in terms of remaining term
i.e.
101
50
- 99
50
= (100+1)
50
- (100 - 1)
50

= (C0.100
50
+ C1100
49
+ C2.100
48
+......)
= (C0100
50
- C1100
49
+ C2100
48
-......)
= 2[C1.100
49
+ C3100
47
+.........]

20. = 2[50.100
49
+ C3100
47
+.........]
= 100
50
+ 2[C3100
47
+............]
> 100
50

=> 101
50
> 99
50
+ 100
50


Example3 :
Find the value of the greatest term in the expansion of
√3(1+(1/√3))
20
.
Solution:
Let Tr+1 be the greatest term, then T r < Tr+1 > Tr+2
Consider : Tr+1 > Tr
=>
20
Cr (1/√3)
r
> 20Cr-1(1/√3)
r-1

=> ((20)!/(20-r)!r!) (1/(√3)
r
) > ((20)!/(21-r)!(r-1)!)
(1/(√3)
r-1
)
=> r < 21/(√3+1)
=> r < 7.686 ......... (i)
Similarly, considering Tr+1 > Tr+2
=> r > 6.69 .......... (ii)
From (i) and (ii), we get
r = 7
Hence greatest term = T 8 = 25840/9
Example 4:
Find the coefficient of x
50
in the expansion of (1+x)
1000
+
2x(1+x)
999
+ 3x
2
(1+x)
998
+...+ 1001x
1000
.
Solution:
Let S = (1 + x)
1000
+ 2x(1+x)
999
+...+ 1000x
999
(1+x) + 1001
x
1000

21. This is an Arithmetic Geometric Series with r = x/(1+x) and d
= 1.
Now (x/(1+x)) S = x(1 + x)
999
+ 2x
2
(1 + x)
998
+...+
1000x
1000
+ 1000x
1001
/(1+x)
Subtracting we get,
(1 - (x/(x+1))) S =(1+x)
1000
+ x(1+x)
999
+...+ x
1000
-
1001x
1000
/(1+x)
or S = (1+x)
1001
+ x(1+x)
1000
+ x
2
(1+x)
999
+...+ x
1000
(1+x)-
1001x
1001

This is G.P. and sum is
S = (1+x)
1002
- x
1002
- 1002x
1001

So the coeff. of x
50
is =
1002
C50
Example 5:
Show that
n
Ck (sin kx) cos (n-k)x = 2
n-1
sin(nx)
Solution:
We have
n
Ck sin kx cos (n-k)x
=1/2
n
Ck [sin (k x + nx - kx) + sin (kx - nx + kx)]
=1/2
n
Ck sin n x + 1/2
n
Ck sin (2kx - nx)
= 1/2 sin n x
n
Ck 1/2 [
n
C0 sin (-nx) +
n
C1 sin (2-
n)x +...
...+
n
Cn-1 sin (n-2)x +
n
Cn sin nx]
= 2
n-1
sin nx + 0 (as terms in bracket, which are equidistant, from
end and beginning will cancel each other.
Example 6:
If (15+6√6)
2n+1
= P, then prove that P(1 - F) = 9
2n+1
(where F
is the fractional part of P).
Solution:
We can write

22. P = (15+6√6)
2n+1
= I + F (Where I is integral and F is the
fractional part of P)
Let F' = (15+6√6)
2n+1


Note: 6√6 = 14.69

=> 0 < 156√6 < 1

=> 0 < (15+6√6)
2n+1
< 1

=> 0 < F' < 1

Now, I + F = C 0 (15)
2n+1
+ C1(15)
2n
6√6 + C2 (15)
2n
(6√6
)
2
+...
F' = C0 (15)
2n+1
- C1(15)
2n
6√6 + C2(15)
2n-1
(6√6 )
2
+...
I + F + F' = 2[C0 (15)
2n+1
+ C2 (15)
2n-1
(6√6 )
2
+...]

Term on R.H.S. is an even integer.
=> I + F + F' = Even integer
=> F + F' = Integer
But, 0 < F < 1 and (F is fraction part)
0 < F' < 1
=> 0 < F + F' < 2
Hence F + F' = 1
F' = (1-f)
.·. P(1-F) = (15 + 6√6 )
2n+1
(15-6√6 )
2n+1

= (9)
2n+1

Example 7:
Using ∫ 0
1
(tx+a-x)
n
dx,prove that ∫ 0
1
x
k
(1-x)
(n-
k)
dx=[
n
Ck (n+1)]
(-1)

Solution:

23. The given integral can easily be evaluated, as follows:
I = ∫0
1
(tx+a-x)
n
dx
= ∫0
1
((t-1)x+1)
n
dx
= [((t-1)x+1)
(n+1)
/((n+1)(t-1))]0
1

=(t
(n+1)
-1)/(n+1)(t-1)=1/(1+n) [1 + t + t
2
+...+ t
k
+...+
t
n
] ...... (i)
Also, I = ∫0
1
(tx+(1-x))
n
dx
=∫0
1

n
Ck (1-x)
(n-K)
t
k
x
k
dx ......
(ii)
Comparing the coefficient of t
k
from (i) and (ii), we get,
∫0
1

n
Ck. x
k
(1-x)
n-k
dx=1/(n+1)
=> ∫0
1
x
k
(1-x)
n-k
dx = 1/(
n
Ck (n+1))
(Hence, proved)

Example 8:
Prove that (-3)
r-1

3n
C2r-1= where k = 3n/2 and n is an even
positive integer.
Solution:
Since n is even integer, let n = 2m,
k = 3n/2 = 6m/2 =3m
The summation becomes,
S =
Now, (1+x)
6m
=
6m
C0 +
6m
C1 x +
6m
C2 x
2
+...
...+
6m
C6m-1 x
6m-1
+
6m
C6m x
6m
...... (i)
(1-x)
6m
=
6m
C0 +
6m
C1 (-x) +
6m
C2 (-x)
2
+...
...+
6m
C6m-1 (-x)
6m-1
+
6m
C6m (-x)
6m
...... (ii)

24. => (1+x)
6m
- (1-x)
6m
= 2[
6m
C1x +
6m
C3x
2
+...+
6m
C6m-1x
6m-
2
]
((1+x)
6m
- (1-x)
6m
)/2x =
6m
C1 +
6m
C3 x
2
+......+
6m
C6m-1
x
6m
- 2
Let x
2
= y
=>((1 + √y)6m -(1-√y)6m)/2√y=
6m
C1+
6m
C3y
+
6m
C3 y+......
6m
C6m-1 y
3m-1

With y = -3, RHS becomes = S.
LHS =

Example9:
If (1+x)
n
= C0 + C1 x + C2 x
2
+...+ Cn x
n
then show that
(i) ∑0≤i<j≤n Ci Cj = 2
2n-1
-
(ii) ∑0≤i<j≤n (Ci + Cj)
2
= (n - 1)
2n
Cn + 2
2n

(iii) ∑0≤i<j≤n ∑Ci Cj = n (2
2n-1
- 1/2
2n
Cn)
Solution:
(i) We have
(C0 + C1 + C2 +...+ Cn)
2
= C0
2
+ C1
2
+...+ Cn
2
+ 2 ∑0≤i<j≤n ∑Ci Cj
we get, (2n)
2
=
2n
Cn + 2 ∑0≤i<j≤n ∑Ci Cj
therefore ∑0≤i<j≤n ∑Ci Cj = 2
2n-1
- (2n)!/2(n!)
2

(Hence, proved)
(ii) ∑0≤i<j≤n ∑Ci Cj n(C0
2
+ C1
2
+ C2
2
+...+ C
n2
) + 2∑0≤i<j≤n ∑Ci Cj

25. = n
2n
Cn + 2{2
2n-1
- (2n)!/2(n!)
2
}
[from part (i)]
= n
2n
Cn + 2
2n
-
2n
Cn
= (n-1)
2n
Cn + 2
2n

(Hence, proved)
(iii) Let ∑0≤i<j≤n ∑Ci Cj
replace i by (n - i) and j by (n - j), we have
P = ∑0≤i<j≤n ∑(2n-i0j) Cn-i Cn-j
= ∑0≤i<j≤n ∑(2n-(i+j)) Ci Cj [·.·
n
Cn =
n
Cn-r]
= 2n ∑0≤i<j≤n ∑Ci Cj - P
.·. ∑0≤i<j≤n ∑(i+j) Ci Cj = n[2
2n-1
- (2n)!/2(n!)
2
]
Example 10. Find the value of (0.98)
-3
upto 2 decimal places.
Solution: (1-0.02)
-3
= [1+(-3)(-0.02)+(-3)(-3-1)/2!(-0.02)
2
+
......] =[1+0.0600+0.0024] 1.0624 = 1.06 { by neglecting
higher power ofmore zeros just after decimal}.