Higher Engineering Mathematics, Sixth Edition

62 Higher Engineering Mathematics

5. Expand(p+ 2 q)^11 as far as the fifth term.
   ⎡
   ⎣

p^11 + 22 p^10 q+ 220 p^9 q^2

+ 1320 p^8 q^3 + 5280 p^7 q^4

⎤

⎦

6. Determine the sixth term of

(

3 p+

q

3

) 13

.

[34749p^8 q^5 ]

7. Determine the middle term of( 2 a− 5 b)^8.
   [700000a^4 b^4 ]


8. Use the binomial theorem to determine, cor-
   rect to 4 decimal places:
   (a)( 1. 003 )^8 (b)( 1. 042 )^7
   [(a) 1.0243 (b) 1.3337]


9. Use the binomial theorem to determine, cor-
   rect to 5 significant figures:
   (a)( 0. 98 )^7 (b)( 2. 01 )^9
   [(a) 0.86813 (b) 535.51]


10. Evaluate( 4. 044 )^6 correct to 3 decimal places.
    [4373.880]

7.4 Further worked problems on the

binomial series

Problem 11.

(a) Expand

1

( 1 + 2 x)^3

in ascending powers ofxas

far as the term inx^3 , using the binomial series.

(b) State the limits ofxfor which the expansion

is valid.

(a) Using the binomial expansion of( 1 +x)n,where

n=−3andxis replaced by 2xgives:

1

( 1 + 2 x)^3

=( 1 + 2 x)−^3

= 1 +(− 3 )( 2 x)+

(− 3 )(− 4 )

2!

( 2 x)^2

+

(− 3 )(− 4 )(− 5 )

3!

( 2 x)^3 +···

= 1 − 6 x+ 24 x^2 − 80 x^3 +···

(b) The expansion is valid provided| 2 x|<1,

i.e. |x|<

1

2

or−

1

2

<x<

1

2

Problem 12.

(a) Expand

1

( 4 −x)^2

in ascending powers ofxas

far as the term inx^3 , using the binomial

theorem.

(b) What are the limits ofxfor which the expan-

sion in (a) is true?

(a)

1

( 4 −x)^2

=

1

[

4

(

1 −

x

4

)] 2 =

1

42

(

1 −

x

4

) 2

=

1

16

(

1 −

x

4

)− 2

Using the expansion of( 1 +x)n

1

( 4 −x)^2

=

1

16

(

1 −

x

4

)− 2

=

1

16

[

1 +(− 2 )

(

−

x

4

)

+

(− 2 )(− 3 )

2!

(

−

x

4

) 2

+

(− 2 )(− 3 )(− 4 )

3!

(

−

x

4

) 3

+···

]

=

1

16

(

1 +

x

2

+

3 x^2

16

+

x^3

16

+···

)

(b) The expansion in (a) is true provided

∣∣

∣

x

4

∣∣

∣<1,

i.e.|x|< 4 or− 4 <x< 4

Problem 13.√ Use the binomial theorem to expand

4 +xin ascending powers ofxto four terms. Give

the limits ofxfor which the expansion is valid.

√

4 +x=

√[

4

(

1 +

x

4

)]

=

√

4

√(

1 +

x

4

)

= 2

(

1 +

x

4

)^12

Using the expansion of( 1 +x)n,

2

(

1 +

x

4

)^1

2

= 2

[

1 +

(

1

2

)(

x

4

)

+

( 1 / 2 )(− 1 / 2 )

2!

(x

4

) 2