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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.
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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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