Binomial Expansion WS3 & Answers

7/29/2019 Binomial Expansion WS3 & Answers

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BINOMIAL EXPANSION

Expansion of ( )+ n a b , n is a positive integer

1. Find, in ascending powers of x, the first 4 terms in the expansion of

(a) 6(2 3 ) x+ (b) ( )5

142

x− (c) ( )8

21 22

x x

−

2. Find the term independent of x in the expansion of

(a)2

912 x

x

+

(b)

6

2

2. x

x

−

3. Find the coefficient of 3 x in the expansion of

(a) ( )5

3 2 x− (b)

12

2

2

. x x

−

4. In the expansion of (2 3 ) ,n x+ the coefficients of 3 4and x x are in the ratio 8:15. Find the value of n.

5. Expand ( )6

24 x+ in ascending powers of x up to and including the term in 3. x Hence find an

approximate value for 6(1.9975) .

6. Expand fully ( )6

2a b+ , simplifying the coefficients.

Hence, or otherwise, write down the term independent of x in the expansion of

( )6

2 2 . x x

+

7. Write down the expansion of ( )5

1 . x+

Hence, by letting 2 , x z z= + find the coefficient of 3

z in the expansion of ( )5

21 z z+ + in powers

of z.

8. Expand 6(1 ) px+ in ascending powers of x, up to and including the term in 2. x

The coefficients of x and 2 x in the expansion of 6(1 )(1 )qx px+ + are 0 and 21− respectively. Given

that 0 and 0, p q> < find the values of the constants p and q.

9. Write down, in ascending powers of x, the first three terms in the expansion of 5(2 ) .ax+ Given that

the first three terms in the expansion of 5 2( 2 ) (2 ) are 96 176 ,b x ax x cx+ + − + find the values of a, b

and c.

10. Given that the coefficients of 2and x x in the expansion of 6 5(1 ) (2 )ax bx+ + are 112 and 80−

respectively, find the integer values of a and b.



2

Expansion of (1 )+ n x

11. Expand the following functions in ascending powers of x up to and including the term in 3. x State

the range of values of x for which expansion is valid.

(a) 3(1 4 ) x−− (b)

52(1 2 ) x+ (c) 4 x+ (d)

13(1 3 ) x

−−

(e) 1

4 9 x−(f) 1

3 2 x−

(g)14

11 x+

(h) ( )2

2 . x−

+

12. Expand 2

1(1 ) x−

, where 1, x < in ascending powers of x, up to and including the term in 3. x

By putting 410 x−= in your expansion, find

21

(0.9999) correct to 12 decimal places.

13. Expand 2(1 ) x−− as a series of ascending powers of x, up to and including the term in 3 , x given that

1. x <

Hence express 21(1 )

x x+−

in the form

2 31 3 . . . , 1, x ax bx x+ + + + <

where the values of a and b are to be stated.

14. Expand 1 x+ as a series of ascending powers of x, up to and including the term in 2, x given that

1. x <

Show that , if x is small, then

2(2 ) 1 , x x a bx− + ≈ +

where the values of a and b are to be stated.

15. Obtain the expansion of 31 3

1

x

x

−

+

in ascending powers of x up to and including the term in 2 . x For

what range of values of x is the expansion valid?

16. Find the first 4 terms in the series expansions of the following functions. In each case, state the values

of x for which the series is convergent.

(a) 1(1 )(1 2 ) x x+ −

(b) 5(1 3 )(1 2 )

x x x+ −

(c)2

2

1

x

x

+

−

(d)5

.(1 3 )(2 )

x

x x

+

+ −

17. Expand 1 2

1 2

x

x

+

−

up to the term in 2 . x By putting 1

100, x = find 51 correct to 3 decimal places.

18. By multiplying the numerator and denominator by 1 , x+ find the expansion of 1

1

x

x

+

−

as far as the

term in 2 . x By putting 1

9, x = prove that 1815 .

81≈



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Binomial Expansion WS3 & Answers