Binomial Theorem

BINOMIAL # 1

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BI NOMI AL D E F I N I T I O N S 1.

AND

R E S U LT S

STATEMENT OF BINOMIAL THEOREM : If x , y  R and n  N, then ; n

(x + y)n = nC0 x n + nC1 x n1 y + nC2 x n2y 2 + ..... + nCr x nr y r + ..... + nCny n =



n

r0

Cr x nr y r .

This theorem can be proved by Induction .

2.

PROPERTIES OF BINOMIAL THEOREM :

(i)

The number of terms in the expansion is (n + 1) .

(ii)

The sum of the indices of x & y in each term is n .

(iii)

The binomial coefficients of the terms nC0 , nC1 .... equidistant from the beginning and the end are equal (  nCr = nCn  r ) .

(iv)

General term : The general term or the (r + 1) th term in the expansion of (x + y)n is Tr+1 = nCr x nr . y r .

(v)

Middle term(s) : (a) (b)

3.

If n is even , there is only one middle term which is given by ; T(n+2)/2 = nCn/2 . x n/2 . y n/2 If n is odd , there are two middle terms which are : T(n+1)/2 & T[(n+1)/2]+1

BINOMIAL THEOREM FOR NEGATIVE OR FRACTIONAL INDICES : If n  Q , then (1 + x) n = 1 + n x +

n (n  1) 2 n (n  1) (n  2) 3 x + x + ........  2! 3!

provided x < 1 .

4.

EXPONENTIAL SERIES :

(i)

x x2 x3 1  1   ex = 1 + 1!  2!  3!  .......  ; where x may be any real or complex & e = Limit n  n

(ii)

ax = 1 +

5.

LOGARITHMIC SERIES :

(i)

2 3 4 ln (1+ x) = x  x  x  x  .......  where 1 < x  1 .

n

x x2 x3 3 lna  l n2a  l n a  .......  where a > 0 . 1! 2! 3!

2

3

2

(ii)

ln (1 x) =  x 

(iii)

ln

4

3

x x x4    .......  2 3 4

where 1  x < 1 .

  x3 x5 (1  x)   ......  x < 1 . = 2 x  3 5 (1  x)  

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BINOMIAL # 2

Note : 1.

Term independent of x  coefficient of x 0 ..

2.

Numerically greatest term in the expansion of (1  x)n , x > 0 , n  N is the same as the greatest term in (1 + x) n .

3.

PROPERTIES OF BINOMIAL COEFFICIENTS :

(i)

C0 + C1 + C2 + ....... + Cn = 2 n

(ii)

C0 + C2 + C4 + ....... = C1 + C3 + C5 + ....... = 2 n1

(iii)

C0² + C1² + C2² + .... + Cn² = 2nCn =

(iv)

C0.Cr + C1.Cr+1 + C2.Cr+2 + ... + Cnr .Cn =

4.

When the index n is a positive integer the number of terms in the expansion of (1 + x)n is finite i.e. (n + 1) & the coefficient of successive terms are : n C0 , nC1 , nC2 , nC3 ..... nCn

5.

When the index is other than a positive integer such as negative integer or fraction, the number of terms in the expansion of (1 + x) n is infinite and the symbol nCr cannot be used to denote the Coefficient of the general term .

6.

If (x < 1 ) . (a) (1 + x)1 = 1  x + x 2  x 3 + x 4  ....  (c) (1 + x)2 = 1  2x + 3x 2  4x 3 + .... 

(2 n) ! n ! n! (2 n) ! (n  r) (n  r) !

(b) (1  x)1 = 1 + x + x 2 + x 3 + x 4 + ....  (d) (1  x)2 = 1 + 2x + 3x 2 + 4x 3 + ..... 

7.

PROPERTIES OF ' e ' :

(a)

e=1+

(b)

' e ' is an irrational number approximately equal to 2.72 ..

(c)

e + e1 = 2  1 

(e)

Logarithms to the base ‘e’ are also called Natural Logarithm .

1 1 1    .......  1! 2! 3!

 

 1 1 1    .......  2! 4 ! 6 ! 

(d)

 

e  e1 = 2  1 

 1 1 1    .......  3! 5! 7! 

EXERCISE I 1.

If ‘a’ be the sum of the odd terms & ‘b’ the sum of the even terms of the expansion of (1 + x)n , then (1  x²)n = (A) a²  b² (B) a² + b² (C) b²  a² (D) none

2.

If the sum of the coefficients in the expansion of (1 + 2x) n is 6561, then the greatest coefficient in the expansion is (A) 1592 (B) 1492 (C) 1792 (D) 1700 9

3.

 3x 2 1  The term independent of x in   is : 3x   2

(A) 27/5 4.

(B) 7/18

(C)  8/81

(D) none of these

Given that the term of the expansion (x 1/3  x 1/2)15 which does not contain x is 5 m where m  N , then m = (A) 1100 (B) 1010 (C) 1001 (D) none

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BINOMIAL # 3

5.

6.

If the second , third and fourth terms in the expansion of (a + b) n are 135, 30 and 10/3 respectively , then : (A) a = 3 (B) b = 1/3 (C) n = 5 (D) n = 7 In the binomial (2 1/3 + 31/3)n, if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is 1/6 , then n = (A) 6 (B) 9 (C) 12 (D) 15

7.

Let R = (6 6 + 14)2n + 1 and f = R  [R] , where [ ] denotes the greatest integer function. Then Rf = (A) 20 2n + 1 (B) 20 2n  1 (C) 20 2n (D) none

8.

The term independent of x in the expansion of  x    x  

is :

(A)  3

(D) 3

4

1  x 

 

(B) 0

1 x

3

(C) 1 2 n

9.

The sum of the coefficients in the expansion of (1  2x + 5x ) is a and the sum of the coefficients in the expansion of (1 + x) 2n is b . Then : (A) a = b (B) a = b 2 (C) a2 = b (D) ab = 1

10.

If k  R and the middle term of ( (A) 3

11. 12.

13.

k + 2)8 is 1120 , then value of k is 2

(C)  3

(B) 2

(D)  4

4

2 12

The coefficient of x in the expansion of (1  x + 2x ) is : (A) 12C3 (B) 13C3 (C) 14C4

(D)

12

C3 + 3 13C3 + 14C3

If (1 + x + x²)25 = a0 + a1x + a2x² + ..... + a50 . x 50 then a0 + a2 + a4 + ..... + a50 is : (A) even (B) odd & of the form 3n (C) odd & of the form (3n  1) (D) odd & of the form (3n + 1) The expression 6

6   2  2 x 2  1  2 x 2  1 +   is a polynomial of degree    2 2  2x  1  2 x  1 

(A) 5 14. 15.

(B) 6

(D) a20 = 2 2 . 3 7 . 7

If x is so small that x 2 and higher powers of x can be neglected , then value of the

(A) 1 +

1  3x  (1  x)5/ 3 is ( 4  x)1/ 2

35 x 24

(B) 1 

35 x 24

(C) 1 +

8 x 9

(D) 1 

8 x 9

C0 C1 C2 C    ......  10 = 1 2 3 11

(A)

17.

(D) 8

If (1 + 2x + 3x 2 )10 = a0 + a1 x + a2 x 2 + .... + a20 x 20 , then (A) a1 = 20 (B) a2 = 210 (C) a4 = 8085

expression

16.

(C) 7

211 11

(B)

211  1 11

x y

y x

The (m + 1)th term of   

(C)

311 11

(D)

311  1 11

2 m 1

is

(A) independent of x (C) dependents on the ratio x/y and m

(B) a constant (D) none of these

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BINOMIAL # 4

18.

If (C0 + C1) (C1 + C2) (C2 + C3) ...... (Cn1 + Cn) = m . C1C2C3 .... Cn1 , then m = (A)

19.

 n  1n 1  n  1 !

(B)

( n 1) n n!

(C)

 n 1n 1 n!

The number of positive terms in the sequence x n = (A) 2

(B) 3

1  195    4  n Pn 

(C) 4

(D)

n n 1  n  1 !

n3 n 1

P3 is : Pn  1

(D) none of these

20.

The numerically greatest terms in the expansion of (2x + 5y)34 when x = 3 & y = 2 is (A) T21 (B) T22 (C) T23 (D) T24

21.

If the second term of the expansion a 1/13  

(A) 4 22. 23. 24.

25.

(B) 3 n

a   is 14a5/2 then the value of a 1 

(C) 12

n

2

10

2

x

10

C0

2/3

(B)

x1 x 1    1/ 3  x  1 x  x1/ 2  10

C7

(D) 5 n  1 (D) 199

3 8

The coefficient of x in the expansion of (1 + x  x ) is (A) 476 (B) 496 (C) 506 

C3 is : C2

(D) 6

The greatest integer less than or equal to ( 2 + 1)6 is (A) 196 (B) 197 (C) 198

In the expansion of 

n n

n1

n

The value of 4 { C1 + 4 . C2 + 4 . C3 + ...... + 4 } is : (A) 0 (B) 5 n + 1 (C) 5 n

(A) 26.

n



(D) 528

10

(C)

, the term which does not contain x is : 10

C4

If x = (7 + 4 3 )2n = [x] + f , then x(1  f) = (A) 2 (B) 0 (C) 1

(D) none (D) 2520

27.

Coefficient of x n  1 in the expansion of , (x + 3)n + (x + 3)n  1 (x + 2) + (x + 3)n  2 (x + 2)2 + ..... + (x + 2)n is (A) n+1 C2 (3) (B) n1 C2 (5) (C) n+1 C2 (5) (D) n C2 (5)

28.

Let (1 + x 2)2 (1 + x)n = A0 + A1 x + A2 x 2 + ...... If A0, A1, A2 are in A.P. then the value of n is : (A) 2 (B) 3 (C) 5 (D) 7

29.

Set of value of r for which, 18Cr  2 + 2 . 18Cr  1 + 18Cr  20C13 contains : (A) 4 elements (B) 5 elements (C) 7 elements (D) 10 elements

30.

If (1 + x + x 2 )n = a0 + a1 x + a2 x 2 +.....+ a2n x 2n then value of a0 + 2a1 + 3a2 + ......+ (2n + 1) a 2 n is n (A) 3 (1 + n) (B) n.3 n (C) 3 n

31.

(D) none

If Cr stands for 4Cr ,then C0 C4  C1 C3 + C2 C2  C3 C1 + C4 C0 = (A) C2

(B) C3

(D) 24

(D) 18

5

32.

The value of the expression (A)

47

C5

(B)

52

47

C5

C4 +

 j1

52  j

C3 is equal to : (C)

52

C4

(D)

49

C4

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BINOMIAL # 5 13

33.

The greatest terms of the expansion (2x + 5y) when x = 10, y = 2 is : (A) 13C5 . 20 8 . 10 5 (B) 13C6 . 20 7 . 10 4 13 9 4 (C) C4 . 20 . 10 (D) none of these

34.

(nC0)2 + (nC1)2 + (nC2)2 + ....... + (nCn)2 = (A) 2 2n (B) 2nCn

35. 36.

(C) (2nCn)2

(D) (n!)2

The remainder when 2 2003 is divided by 17 is (A) 1 (B) 2 (C) 8

(D) none of these

If n is a natural number which is not a multiple of 3 and (1 + x + x 2 )n = 2n

n

r0

r0

 a r x r , then value of  ( 1) r (a r )( n C r ) is (A)  1

(B) 2

(D) (1)n

(C) 0

EXERCISE II 1.

Show that the coefficient of x 10 in the expansion of (1 + x 2  x 3)9 is 882 .

2.

Show that the term independent of x in the expansion of 1  x  

 

6 x

10

is ,

5

1+ 

10

r 1

C2 r

2r

Cr 6 r .



3.

Show that there are 32 integer terms in the expansion of ,

4.

Find numerically the greatest term in the expansion of : (2 + 3x)9 when x =

(i)

5.

3 2

(ii)

 In the binomial expansion of ,  y  

  4 ( 2 y ) 

345

124



.

1 5

(3  5x)15 when x =

n

1

the first three coefficient form an A.P.

in the order . Find other terms in the expansion of which the power of ' y ' is a natural number . 2

6.

Given s n= 1 + q + q² + ..... + q that

n+1

C1 +

n+1

C2.s 1 +

n+1

n

n

 q  1  q  1 q 1 & Sn = 1 + +  2  + .... +  2  , q  1 , prove 2

C3.s 2 +....+

n+1

Cn+1.s n = 2 n . Sn .

1 1. 3 1. 3 . 5 1. 3 . 5 . 7     ........  3 3. 6 3. 6 . 9 3. 6 . 9 .12

then prove that x² + 2x  2 = 0 .

7.

If x =

8.

Given that , (1 + x + x 2 + x 3)5 = a0 + a1 x + a2 x 2 + ...... + a15 x 15 . Find a10 .

9.

Prove that

10.

Show that , lnx =

11.

If 3 3  5





(72)!

 36! 2

2n  1

 1 is divisible by 73 . x  1 1 x2  1 1 x3  1  .  . 2 x  1 2 (x  1) 3 (x  1) 3

+...... (x > 0)

= p+f where p is an integer and f is a proper fraction then find the

value of f (p+f) . n N . IIT-ian’s PACE ; ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com

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BINOMIAL # 6

12.

p

  Prove that if ' p ' is a prime number greater than 2, then the difference  2  5   2 p+1  





is divisible by p, where [ ] denotes greatest integer . 13.

If ' n ' & ' r ' are coprime , prove that nCr is divisible by n .

14.

If C0 , C1 , C2 , ..... , Cn are the combinatorial coefficients in the expansion of (1 + x) n, n  N , then prove that , ( n  1) (2 n)! n! n!

1 . Co² + 3 . C1² + 5 . C2² + ..... + (2n+1) Cn² = 15.

If n is an integer greater than 1 , show that ; a  nC1(a1) + nC2(a2)  ..... + (1)n (a  n) = 0

16. 17.

C1  C 2  C 3  ...... 

C n  2n  1 

n 1 2

If (1+x)n = C0 + C1x + C2x² + .... + Cn xn , then show that the sum of the products of the Ci’s taken two at a time , represented by

  Ci C j 2 n! is equal to 2 2n1  . 0 i j n 2 (n !) 2

18.

If (1 + 2 x + 2 x 2)n = k 0 + k 1 x + k 2 x 2 + ...... + k 2 n x 2 n . Prove that , k 2 = 4 n 2  2 2  n n ( 1 + 3 . n  1C2 + 5 . n  1C3 + ...... + (2 n  1) n  1Cn  1) .

19.

If (1 + x)n =

n

C

r

. x r then prove that ;

r0

2 2 . C0 2 3 . C1 2 4 . C 2 2n  2 . C n 3n  2  2n  5    ......   1. 2 2.3 3. 4 (n  1) (n  2) ( n  1) (n  2)

20.

Prove that the sum to (n+1) terms of

C0 C1 C2    ......... equals n (n  1) (n  1) ( n  2) ( n  2) (n  3)

1



x n1. (1 x)n+1 . dx & evaluate the integral .

0

3n

21.

Prove that ,

 k  0

6n

C2 k  1 ( 3)k = 0 .

2 12 28 50 78     + ...... = 5 e + 2 1! 2! 3! 4! 5!

22.

Prove that ,

23.

If the series 1 +

x3 x6  + ... 3! 6!

; x+

x4 x7  + ... 4 ! 7!

;

x2 x5 x8   + ... are denoted 2! 5! 8!

by a , b , c respectively, show that : a 3 + b 3 + c3  3abc = 1 . 24.

Prove that the coefficient of x n in the expansion of log e (1 + x + x²) is 

2 n

or

1 n

according as n is/or is not a multiple of 3 . 25.

Find the sum of the following infinite series ,

26.

Prove that ,

x2 2 x3 3x4 4 x5     ....... given x < 1 2 3 4 5

1 1 1 1 1 1 1    ... =     .... = ln 2 1. 2 3 . 4 5 . 6 2 1. 2 . 3 3 . 4 . 5 5 . 6 . 7

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BINOMIAL # 7

27.

If  ,  are the roots of the equation ax² + bx + c = 0 , then show that : 

log e (a + bx + cx²) = log e a  (  ) x  

 2  2 2  3  3 3 x  x  ....... 2 3 

1 1 1   n  1 2 ( n  1) 2 3 (n  1) 3

 .....

Show that :

29.

Prove that , 1. 2 . 3  3. 4 . 5  5 . 6 . 7 +........ = 3 ln2  1

30.

If x denotes 2  3 , n N & [x] the integral part of x then find the value of ;

5

7





+ .... =

1 1 1   n 2 n2 3 n3

28.

9

n

x  x² + x[x] . 31.

32.

Find the cube root of 1001 correct to 5 decimal places without using calculator or tables . Also evaluate (0.99) 15 correct to 4 places . x

2

n

  Find the index ' n ' of the binomial    if the 9th term of the expansion has numerically  5 5

the greatest coefficient (n  N) . If C0 , C1 , C2 , ..... , Cn are the combinatorial coefficients in the expansion of (1 + x)n , n  N , then prove the following : 33.

Co²  C1² + C2²  C3² + ...... + (1)n Cn² = 0 or (1)n/2 Cn/2 according as n is odd or even .

34.

(n1)² . C1 + (n3)² . C3 + (n5)² . C5 +..... = n (n + 1)2 n3 .

35.

If a0 , a1 , a2 , ..... be the coefficients in the expansion of (1 + x + x²) n in ascending powers of x , then prove that : a0 a1  a1 a2 + a2 a3  .... = 0 (ii) a0a2  a1a3 + a2a4  ..... + a2n  2 a2n = an + 1 .

(i) (iii)

E1 = E2 = E3 = 3 n1 ; where E1= a0 + a3 + a6 + ........ ; E2 = a1 + a4 + a7 + ........ & E3 = a2 + a5 + a8 + .......

36.

Prove that : C 0  C1  C 2  C 3  ........(1) n C n 

37.

If (1+x)n = C0 + C1x + C2x² + ..... + Cn x n , then show that :

1

C1(1x) 

5

9

13

4n 1

4 n . n! . 1. 5 . 9 .13..... ( 4 n  3) ( 4 n  1)

C2 C 1 1 1 (1x)² + 3 (1x)3 ....+ (1)n1 (1x)n = (1x) + (1x²) + (1x 3) 2 3 n 2 3

1 n

+......+ (1x n) 1n 2 3 4 1 ( 1) n  1 n n C1 nC2+ nC3 nC4 + ..... + . Cn= . 2 3 4 5 n 1 n 1

38.

Prove that ,

39.

Prove that , (2nC1)²+ 2 . (2nC2)² + 3 . (2nC3)² + ... + 2n . (2nC2n)² =

40.

If (1 + x + x 2)n =

(4 n  1)!

(2 n  1) !

2

2n

 ar xr , n  N , then prove that

r0

(r + 1) ar + 1 = (n  r) ar + (2 n  r + 1) ar  1 .

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BINOMIAL # 8

EXERCISE 1.

III

Find the coefficient of x 50 in the expression : (1 + x)1000 + 2x . (1 + x)999 + 3x² (1 + x)998 + ..... + 1001 x 1000 [ REE ’90 , 6 ] [ REE ’91 , 6 + 6 ]

2.

 C   k  C k   k 1  k 1 n

(a)

If n is a positive integer & C k = nCk , find the value of

(b)

Find the sum of the following series upto infinity . 1

2 1



2 2

.

3  2 2 5 2  7 17  12 2    ......... 12 80 24 2

3.

[ JEE ’92 , 6 + 2 ] 2n

(a)

2

3

If

a

2n r

(x  2) r 

r0

b

r

r0

(x  3) r & a = 1 for all k  n, then show that b = k n

2n+1

Cn+1.

(b)

The expression [x + (x 31)1/2]5 + [x  (x 31)1/2]5 is a polynomial of degree : (A) 5 (B) 6 (C) 7 (D) 8

4. (a)

[ REE '92 , 6 + 2 ] Determine the value of ' x ' in the expression (x + x ) , if the third term in the expression is 10,00,000 . Where t = log 10 x .

(b)

Sum the following series : 9 

t 5

16 27 42    ........  2! 3! 4!

5.

[ REE '93 , 6 + 6 ]  log10  3   1/ 2  x  2 log 3 2   2   x

(a)

Find the value of ' x ' for which the sixth term of



1/ 5



  

m

is equal to

21 & binomial coefficients of second , third & fourth terms are the first , third & fifth terms of an arithmetic progression . [ Take every where base of log as 10 ] 

1

a2 

1

a3 

1

(b)

2 4 6 Find the sum of a  x  2   2  x  4   3  x  6   ........ and determine the values x x x of a & x for which it is valid .

6.

Let n be a positive integer . If the coefficients of 2nd , 3rd & 4th terms in the expansion of (1+x)n are in AP , then the value of n is ______ . [ JEE ’94 , 2 ]

7.

Given that the 4th term in the expansion of  2  8  has the maximum numerical



3x 

10

value , find the range of values of ' x ' for which this will be true . [ REE ’94 , 6 ] 8.

If a0 , a1 , a2 , ..... be the coefficients in the expansion of (1+x+x²) n in ascending powers of x , then prove that a 0² a1² + a2²  a3² + ..... + a2n² = an . [ JEE ’94 , 5 ]

9.

Find the sum of the infinite series n

 k 1

2k  1 k !( n  k )!

a 1 + a2 + a3 + .......... , where a n = (log e 3)n

for each positive integer n .

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BINOMIAL # 9

[ REE ’95 , 6 ] n4

Let (1+x²)² . (1+x)n =

10.

a K 0

K

. x K . If a1 , a2 & a3 are in AP, find n .

[ REE ’96 , 6 ] In the expansion of the expression (x + a) , if the eleventh term is the geometric mean of the eighth and twelfth terms , which term in the expansion is the greatest ? [ REE ’96 , 6 ] n th th In the binomial expansion of (a  b) , n 5, the sum of the 5 and 6 terms is zero. Then a/b equals : 15

11.

12.

(A)

n5 6

(B)

5

n4 5

6

(C) n  4

(D) n  5 [ JEE '2001 , 1 ]

ANSWER SHEET

EXERCISE I 1. A

2. C

3. B

4. C

5. ABC

6. B

7. A

8. B

9. A

10. B

11. D

12. A

13. B

14. ABC

15. B

16. B

17. C

18. B

19. C

20. B

21. A

22. D

23. B

24. A

25. C

26. C

27. C

28. AB

29. C

30. A

31. A

32. C

33. C

34. B

35. C

36. C

EXERCISE II 4. (i) T7 =

7.313 2

(ii) 455 x 3 12

5. If n = 4 , then T1 = y 2 . If n = 8 , then T1 = y 4 , T7 = 11. 2 2n+1

8. 101

25. ln (1  x) +

x 1 x

35 y 8

30. 1

31. 10.00333 ; 0.8601

32. n = 12

EXERCISE III 1.

n (n  1) 2 ( n  2) 12

6. S = ln  1  

9. 6 ln (27e)

2. 2 

a   a x2  a 2   x2

1 2 ln 2 2

4. x = 10 or 10 5/2

where 1 < a < 1 & x > 1

10. n = 3 or 4

11. T8

12. 4e  3

5. x = 2 or 0 64   64   ,  2   2 ,    21 21 

7. x    14.

12

C6

15. B

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