Chapter 3 | The Binomial Theorem Flipbook PDF

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Chapter (3)

The Binomial Theorem Chapter (3) The Binomial Theorem

Binomial Expression An expression of the form (x + y), two terms and only two terms, raised to any power is called a binomial. For examples, a + b, 2x + 3y, 4p + 5q, x + Binomial Expansion Binomial

1 a x2 x ,  , (x + y)5, (1 + )–3, etc., x x b a

Expansion

Power

(x + y)

1

0

1

(x + y)1

x+y

1

1 1

2

1 2 1

3

1 3 3 1 1 4 6 4 1

0

2

x + 2xy + y

2

(x + y)

3

x + 3x y + 3xy + y

(x + y)4

x4 + 4x3y + 6x2y2 + 4xy3 + y4

4

(x + y)5

x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + y5

5

(x + y)

3

2

2

2

3

Coefficients

1 5 10 10 5 1

Important Features in Binomial Expansion 1. power of binomial = n, no of terms = n + 1 2. power of x + power of y = power of binomial (in every terms) 3. power of x  descending, power of y  ascending 4. Coefficient Pattern  Each coefficient of a line is obtained by adding the two coefficients on either side of it in the line above. This coefficient pattern is called Pascal’s Triangle. 1|Page

Chapter (3)

The Binomial Theorem

The Binomial Theorem The expansion of (x + y)n can be expressed as n

n

n 0

n

n–1

(x + y) = C0 x y + C1 x

1

n

n–2

y + C2 x

2

n

n–r

r

n

y + … + Cr x y + … + Cn – 1 x y

n–1

n

0 n

+ Cn x y .

This is known as the Binomial Theorem. n

n

n

2

n

r

n

n

1.

Special Case: (1 + x) = 1 + C1 x + C2 x + … + Cr x + … + Cn – 1 + x .

2.

This theorem is true for all values of x and y and when n is a positive integer.

3.

(r + 1)th term or general term = nCr xn – r y r where r is a whole number.

4.

The binomial coefficients are all integers.

5.

The coefficient of terms equidistance from the beginning and end of the expansion are equal. That is … (i) nC0 = nCn

(ii) nC1 = nCn – 1

(iii) nCr = nCn – r

6.

n

C0 + nC1 + nC2 + … + nCn = 2n (prove yourself)

7.

n

8.

The term independent of x = the term with no power of x = the constant term

n

n

n

n

n

n–1

C0 + C2 + C4 +… = C1 + C3 + C5 + … = 2

(prove yourself)

The Middle Term in the Expansion of (x + y)n When n is even there is a single middle term. Middle term = (

n + 1)th term (prove yourself) 2

When n is odd there are two middle terms. Middle term = (

n  1 th n  3 th ) term and ( ) term (prove yourself) 2 2

2|Page

Chapter (3)

The Binomial Theorem

Exercise (1) 1.

Expand the following. (a) (1 + 2x)4

(b) (2x – y)5

(c) (a + 2b)3

x 5 ) 2 1 (g) (x – )3 y

1 1 3 (e) ( x + y)4 (f) (1 – )4 2 3 x a 3 2 (h) ( – )5 (i) (3y – )3 2 b x 1 3 a 3b 2 5 (j) (2xy – )6 (k) (3x + )3 (l) ( – ) y x 2b a 3 Find in ascending power of x, the first three terms in the expansion of

(d) (2 +

2.

5

4

(i) (1 + 2x)

2

(ii) (3 – x) . Hence find the coefficient of x in the expansion

of (1 + 2x)5 (3 – x)4. 3.

4.

Find in ascending power of x, the first three terms in the expansion of 1 4 4 2 (i) (1 + 2x) (ii) (2 – x) . Hence find the coefficient of x in the expansion 2 1 4 4 of (1 + 2x) (2 – x) . 2 Find in ascending power of x, the first three terms in the expansion of 4

2 5

2

(i) (1 – 2x) and (2 + x ) . Hence find the coefficient of x in the expansion of 4

2 5

(1 – 2x) (2 + x ) . 5.

Find in ascending power of x, the first three terms in the expansion of 5

(i) (1 + 2 x)

5

2

(ii) (3 – x) . Hence obtain the coefficient of x in the expansion 2 5

of (3 + 5x – 2x ) . 6.

Expand, in descending power of x, the expansion of (2x +

1 5 1 5 ) and (2x – ). 2x 2x

1 5 1 5 ) + (2x – ) 2x 2x 1 5 1 5 (ii) find the coefficient of x2 in the expansion of (2x + ) (2x – ). 2x 2x 3|Page

Hence, or otherwise (i) simplify (2x +

Chapter (3)

The Binomial Theorem

7.

Evaluate ( 2  1)6  ( 2  1)6 .

8.

Simplify (2 – 3 ) + (2 + 3 ) .

9.

Simplify (x + x  1 )6+ (x – x  1 )6.

7

7

Exercise (2) 1.

Find and simplify the coefficient of x 7 in the expansion of (x2 –

2.

Find the coefficient of x 6 in the expansion of (a) (2 + x)8

3.

Find the coefficient of x – 10 in the expansion of (2 –

2 8 ). x

(b) (x –

1 14 ) . x2

3 14 ) . x

8.

1 20 4 Find the term in x and the term independent of x in the expansion of (x + ) . x 2 6 2 Find the term independent of x in the expansion of (x + ) . x 1 Find the term independent of x in the expansion of ( 2 – x)9. 2x 2 x Find the 4th term in the expansion of ( + 2x)7. 2 Find the middle term in the expansion of (x2+ 2y)10.

9.

Find the two middle terms in the expansion of (a +

10.

1 Find the constant term in the expansion of ( –2x2)9. x

11.

Find the coefficient of x4 in the expansion of (x2 – 5x + 12) (x –

4. 5. 6. 7.

12. 13. 14.

1 11 ) . b

2 6 ). x 5 5 Find the term independent of x in the expansion of (5x – )6(5x + )6. x x x Find the coefficient of x 3 in the expansion of (1 + 2x)5 – (1 – )8. 2 1 Find the middle term and the constant term in the expansion of (2x2 – )12. 2x

4|Page

Chapter (3)

The Binomial Theorem

15.

2

16. 17. 18. 19.

Using binomial theorem, find the coefficient of x in the expansion of 2 5 (3 + x – 2x ) . 1 12 Find the middle term and the term independent of x in the expansion of (x – ) . x 3 1 9 3 Find the the term independent of x in the expansion of (1 + x + 2x ) ( x 2  ) . 2 3x

x 3  2 )10. 3 x 4 2 5 Find the coefficient of x in the expansion of (1 + 2x + x ) . Find the the term independent of x in the expansion of (

Exercise (3) 4

4

1.

Expand (a + b) and by choosing the appropriate values of a and b compute 99 .

2.

Write down and simplify the first four terms in the binomial expansion of 6

6

(1+ 2x) . Use it to find the value of 1.02 , correct to three decimal places. 7

3.

Using binomial theorem, find the value of 1.01 to three decimal places.

4.

Write down and simplify the first four terms in the binomial expansion of (1 – 2x)5. Use it to find the value of 0.985 to four decimal places. 5

5.

Use binomial theorem to find the value of 1.9 to four decimal places.

6.

Obtain the first four terms of the expansion of (1 + p) in ascending power of

6

p. By substituting p = x + x2, obtain the expansion of (1 + x + x2)6 as far as the term in x2. Hence find the value of 1.01016 to three decimal places. 5

7.

Use binomial theorem to find the value of 1.0404 to three decimal places.

8.

Using binomial theorem, determine which is larger 1.24000 or 800.

9.

Find the first 4 terms of the expansion of (1 +

x 10 ) in ascending powers of x, 2 giving each term in its simplest form. Use your expansion to estimate the value

of (1.005)10, giving your answer to 5 decimal places. 5|Page

Chapter (3)

The Binomial Theorem

Exercise (4) 1.

In the binomial expansion of (3 + kx)9, the coefficient of x3 and x 4 are equal. Calculate the value of k.

2.

4

5

10

If the coefficients of x and x in the expansion of (3 + kx) are equal, find the value of k.

3.

If the coefficient of x 4 in the expansion of (3 + 2x)6 is equal to the coefficient of x 4 in the expansion of (k + 3x)6, find k.

4.

2

6

The coefficient of x in the expansion of (2x + k) is equal to the coefficient 5

8

of x in the expansion of (2 + kx) . Find k. 5. 6.

a In the expansion of (x2 + )8, a ≠ 0, the coefficient of x7 is four times the x coefficient of x10. Find the value of a. 3

5

6

Given that the coefficient of x in the expansion of (a + x) + (1 – 2x) is –120, calculate the possible values of a.

7.

If the 2nd and 3rd term in (a + b)n are in the same ratio as the 3rd and 4th in (a + b)n + 3, then find n.

8.

Evaluate the coefficients of x4 and x5 in the expansion of ( evaluate the coefficient x5 in the expansion of (

9.

10. 11.

x – 3)7. Hence 3

x – 3)7(x + 6). 3

1 In the binomial expansion of (1 + )n, the 3rd term is twice the 4th term. Calculate 4 the value of n. 3 The 3rd term in the expansion of (1 + )n is 6 times the 2nd term. Calculate the 5 value of n. In the expansion of (2 + 3x)n, the coefficient of x3 and x4 are in the ratio 8 : 15. Find the value of n. 6|Page

Chapter (3) 12.

13.

The Binomial Theorem

1 In the expansion of (1 + )n, the third term and the fourth term are in the ratio 3 3 : 2. Find the value of n and the middle term of that expansion. k 10 –6 In the expansion of (x + ) , the coefficient of x and the term independent x of x are in the ratio 10 : 7. Find the value of k .

14.

4

6

The ratio of the coefficient of x in the expansion of (3 – 2x) and the coefficient of x4 in the expansion of (k + 2x)7 is 1 : 7. Find the value of k.

a 7 5 ) , x  0, the coefficient of x is 20 times the x 11 coefficient of x . Find the possible values of a. 2

15.

In the expansion of (x –

16.

In the expansion of (3 + 4x)n, the coefficients of x4 and x5 are in the ratio 3 : 4. 5

6

Find the value of n. Calculate the ratio of the coefficients of x and x . 17.

1 9 9 8 7 6 ) is 512x + 576 x + ax + bx + … . 4 Calculate (i) the values of a and b. (ii) the coefficient of x7 when the expansion The binomial expansion of (2x +

of (2x +

1 9 ) is multiplied by 4x – 1. 4

Exercise (5) 1.

Given that the coefficient of x2 in the expansion of (4 + kx) (2 – x)6 is zero, find the value of k.

2.

Given that the coefficient of x2 in the expansion of (1 – ax)6 is 60 and a > 0, find the value of a.

3.

Given that the coefficient of x2 in the expansion of (1 – 3x) (1 + ax)6 and that a is positive, evaluate the value of a. 7|Page

Chapter (3) 4.

5.

The Binomial Theorem 2

3

Given that the coefficients of x and x in the expansion of (3 + x)20 are a and a b respectively, evaluate . b If the coefficient of x2 in the expansion of (4 + kx) (2 – x)6 is 384, find the value of k and the coefficient of in that expansion.

6.

Find, in ascending powers of x, the first three terms of (1 – 4x) (1 + kx)5. If 2

the coefficient of x is 16, find the value of k, and the coefficient of x . 7.

If, in the expansion of (1 + x)m(1 – x)n, the coefficient of x and x2 are –5 and 7 respectively, then find the value of m and n.

8. 9.

1 5 If the 4th term in the expansion of (ax + )n is , then find the value of a and n. x 2 2 2 3 n The expansion of (1  px ) where n > 0, is 1 + 20x + 45p x + kx + … . Calculate the values of n, p and k.

10.

The expansion of (1 + px + qx2)8 is 1+ 8x + 52x2 + kx3+… . Calculate the values of p, q and k.

11.

Write down the third and fourth terms in the expansion of (a + bx)n. If these terms are equal, find the value of a in terms of n, b and x.

12.

It is known that a, b and n are positive integers and the first three terms in the binomial expansion of (a + b)n are 729, 2916, 4860 respectively. Find a, b and n.

13.

2

In the expansion of (x –

1 n th ) in descending power of x, the 5 term is 2x

independent of x. Find the value of n and the 4th term. 14.

nd

rd

n

rd

th

If the 2 and 3 terms in (a + b) are in the same ratio as the 3 and the 4 terms in (a + b)n + 3 , find the value of n .Calculate also the middle term of (a + b)n + 3. 8|Page

Chapter (3) 15.

The Binomial Theorem

The coefficients of three successive terms in the expansion of (1 + x)n are 165, 330 and 462 respectively, find the value of n.

16.

th

th

Find the value of r if the coefficients of (2r + 4) and (r – 2) terms in the expansion of (1 + x)18 are equal.

17.

The coefficients of the (r – 1)th, rth and (r + 1)th terms in the expansion of (x + 1)n are in the ratio 1 : 3 : 5. Find n and r.

18.

If the 21st and 22nd terms in the expansion of (1 + a)44 are equal, then find the value of a.

19.

11

3

In the expansion of (1 + 2x) , the coefficient of x is k times the coefficient 2

of x . Find the value of k. 20.

If the coefficients of three consecutive terms of (1 + x)

n+5

are in the ratio

5 : 10 : 14, find the value of n. 21.

3

4

2

If the coefficients of x and x in the expansion of (1+ ax + bx ) ( 1− 2x) are both zero, find the value of a and b.

22.

Given that the expansion of (a + x) (1 – 2x)n in ascending powers of x is 3 – 41x + bx2 + …, find the values of the constants a, n and b.

23.

Given that (3 + x)5 + (3 – x)5 = p + qx2 + rx4, find the value of p, of q and of r.

24.

Write down the first three terms in the expansion, in ascending powers of x, of (1 + px)5, where p is a constant. The first three terms in the expansion of (1 + qx) (1 + px)5, where p and q are integers are 1 – 10x + 15x2. Obtain two equations in p and q and hence find their values.

25.

If the constant term in the expansion of

 x2 3 x 2  

8

k   is 16128, find k. x 9|Page

Chapter (3)

The Binomial Theorem

Exercise (6) 1.

Expand (1 + x)n and hence show that nC0 + nC1 + nC2 + … + nCn = 2n.

2.

If the sum of all coefficients in the expansion of (1 + 2x)n is equal to the sum of all coefficients in the expansion of (2 – kx)n, show that k = – 1 or 5.

3.

In the binomial expansion of (1 + x)

m+n

m

n

, prove that the coefficients of x and x

are equal. 4.

Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2n – 1.

5.

2n

Prove that the coefficient of one middle term in the expansion of (1 + x) is equal 2n-1

to the sum of coefficients of two middle terms in the expansion of (1 + x) . 6.

Prove that the sum of the coefficients of xr and xr – 1 in the expansion of (1 + x)n r

n+1

is equal to the coefficient of x in the expansion of (1 + x) 7.

n

th

.

th

If in the expansion of (a + b) , the coefficients of p and q terms are equal, show that p + q = n + 2, where p ≠ q.

8.

If the coefficient of (m + 1)th term in the expansion of (1 + x)2n is equal to that th

of (m + 3) term, then show that m = n − 1. 9. 10.

11.

7

−7

ax 

1 bx )11 are equal, prove

If the coefficients of x and x in the expansion of ( that ab = 1. If a1, a2, a3 and a4 are the coefficients of any four consecutive terms in the a3 a1 2 a2 expansion of (1 + x)n, show that .   a1  a2 a3  a4 a2  a3 If the coefficients of ak − 1, ak, ak + 1 in the binomial expansion of (1 + a)n are in A.P., then prove that n2− n (4k + 1) + 4k2 – 2 = 0. 10 | P a g e

Chapter (3) 12.

The Binomial Theorem

If the coefficient of the middle term in the expansion of (1+x)2n+2 is p and the coefficients of middle terms in the expansion of (1+x)2n+1 are q and r, show that p = q + r.

13.

In the expansion of (x + a)n, the sum of odd terms is P and sum of even terms is Q, show that P2 – Q2 = (x2 − a2)n.

14.

In the expansion of (x + a)n, the sum of odd terms is A and sum of even terms is B, show that (x + a)2n − (x − a)2n = 4AB.

15.

Write down the third and fourth terms in the expansion of (a + bx)n. If these terms are equal, show that 3a = (n – 2) bx.

16.

n

The first four terms in the binomial expansion of (a + b) , in descending powers of a are w, x, y and z respectively. Show that (n – 2) xy = 3nwz.

17.

n

The first three terms in the binomial expansion of (a + b) , in ascending powers of b are p, q and r respectively. Show that

18.

2n q2 = . pr n  1

2

3

It is given that the coefficient of x is equal to the coefficient of x in the binomial expansion of (2k + x)n, where k is a constant and n is a positive integer. Prove that n = 6k + 2.

19.

2

n

If the sum of all coefficients in the expansion of (9x – 1) is a and the sum 2 n

3

of all coefficients in the expansion of (1 + x ) is b, show that a – b = 0. 20.

If the coefficient of (p + 1)th term in the expansion of (1 + x)2n is equal to th

that of the (p + 3) term, show that p = n – 1. 21.

Using binomial theorem, prove that 9950 + 10050 < 10150..

11 | P a g e

Chapter (3)

The Binomial Theorem

Exercise (7) 1.

If n  N, show that 4n– 3n – 1 is divisible by 9 by using binomial theorem.

2.

Using binomial theorem, show that 9 – 8n – 1 where n  N is divisible by 64.

3.

Using binomial theorem, show that 24n + 4 – 15n – 16, where n  N is divisible

n

by 225. 2n + 2

4.

Using binomial theorem, prove that 3

− 8n − 9 is divisible by 16.

5.

Using binomial theorem, prove that 6 – 35n – 1 is divisible by 1225 where

2n

n is a positive integer and n  2. 6.

n

n

Using binomial theorem, show that 24 – 2 (7n + 1) is a multiple of 196, where n is a positive integer.

7.

Two positive integers, a and b are given by a = b  1. By using binomial expansion, show that for all positive integral values of n the expression 2n

2

a + 2nb  1 is exactly divisible by b . By choosing suitable values for a and n, show that 240 + 119 is divisible by 9. 8.

Using binomial theorem, show that 34n + 1 + 16n  3 is divisible by 256 if n is a positive integer.

9.

Using binomial theorem, prove that 6n  5n always leaves the remainder 1 when divided by 25 for all positive integers n.

10.

If a and b are distinct integers, prove that an  bn is divisible by a  b, whenever n is a positive integer by using binomial theorem.

12 | P a g e