Cambridge Additional Mathematics

274 Counting and the binomial expansion (Chapter 10)

Example 17 Self Tutor

Write down the first three and last two terms of the expansion of

³

2 x+^1

x

́ 12

.

Do not simplify your answer.

³

2 x+

1

x

́ 12

=(2x)^12 +

¡ 12

1

¢

(2x)^11

³ 1

x

́ 1

+

¡ 12

2

¢

(2x)^10

³ 1

x

́ 2

+::::

::::+

¡ 12

11

¢

(2x)^1

³ 1

x

́ 11

+

³ 1

x

́ 12

Example 18 Self Tutor

Find the 7 th term of

³

3 x¡

4

x^2

́ 14

. Do not simplify your answer.

a=(3x), b=

³

¡ 4

x^2

́

, and n=14

Given the general term Tr+1=

¡n

r

¢

an¡rbr, we let r=6

) T 7 =

¡ 14

6

¢

(3x)^8

³¡ 4

x^2

́ 6

Example 19 Self Tutor

In the expansion of

³

x^2 +

4

x

́ 12

, find:

a the coefficient ofx^6 b the constant term.

a=(x^2 ), b=

³

4

x

́

, and n=12

) the general term Tr+1=

¡ 12

r

¢

(x^2 )^12 ¡r

³ 4

x

́r

=

¡ 12

r

¢

x^24 ¡^2 r£

4 r

xr

=

¡ 12

r

¢

4 rx^24 ¡^3 r

a If 24 ¡ 3 r=6

then 3 r=18

) r=6

) T 7 =

¡ 12

6

¢

46 x^6

) the coefficient ofx^6 is

¡ 12

6

¢

46 or 3 784 704.

b If 24 ¡ 3 r=0

then 3 r=24

) r=8

) T 9 =

¡ 12

8

¢

48 x^0

) the constant term is

¡ 12

8

¢

48 or 32 440 320.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_10\274CamAdd_10.cdr Monday, 6 January 2014 12:02:27 PM BRIAN