Page 39 - PRE-U STPM MATHEMATICS (T) TERM 1

Mathematics Term 1 STPM Chapter 2 Sequences and Series

Example 37

2 7
Expand (1 + x + 2x ) up to the term in x .
3
2
Solution: Rewriting (1 + x + 2x ) as a binomial,
2
2 7
(1 + x + 2x ) = [1 + (x + 2x )] 7
2 3
= 1 + 7(x + 2x ) + 7 · 6 (x + 2x ) + 7 · 6 · 5 (x + 2x ) + …
2 2
2
1 · 2 1 · 2 · 3
We stop the expansion here
2 4
because the next term (x + 2x )
4
contains x and higher powers of x.
2 = 1 + 7(x + 2x ) + 21(x + 4x ) + 35x + …
3
2
2
3
3
2
= 1 + 7x + 35x + 119x + … Ignore x and higher powers of x.
4
Exercise 2.7
1. Use the binomial theorem to expand
2 6
7
4
(a) (p + q) (b) (m – n) (c) (2 + k )
1

(d) (2a – b) 5 (e) 1 x + 1 2 3 (f) 1 y – 2y 2 8
x
2. In each of the following expansions, find the term as stated.
th
10
8
(a) (1 + x) , 5 term (b) (2 – 3x) , term in x 2
2 7
4 6
12
th
(c) (2a + b) , 10 term (d) (p – 3q ) , term in p q
(e) 1 x – 1 2 6 , constant term (f) 1 x + 1 2 9 , term in x 1 3 .
2
x
x
3. Expand the expression
1
1 2x + 1 2 2 5 + 2x – 1 2 2 5 ,
simplifying the terms. x x
n
2
4. The coefficient of x is four times the coefficient of x in the expansion of (1 + x) . Find the value of n.
3
n
2
1
5. In the binomial expansion of 1 + 1 x , the coefficients of the fourth and fifth terms are equal. Find the
value of n. 3
8
5
4
6. The coefficient of x in the binomial expansion of (1 + 5x) is the same as the coefficient of x in the
7
expansion of (a + 5x) . Find the value of a.
n
7. If the first three terms in the expansion of (1 + ax) in ascending powers of x are 1 – 4x + 7x , find n
2
and a.
8. Find the first four terms of each of the following expansions, in ascending powers of x.
2 7
(a) (1 + x) (b) (1 + x – x )
7
3
2 8
9. Expand (1 + 2x + 3x ) in ascending powers of x up to and including the term in x .
6
10. Find the first three terms, in ascending powers of x, of the expansion (1 – 3x)(1 + 2x) .
11. Find the coefficient of the terms in x as indicated, in the following expansions.
2
(a) (1 + x )(2 – 3x) , term in x (b) (1 – 3x – 2x )(1 + x ) , term in x 20
2 20
3
7
2
1
5
(c) x x – x 2 2 2 12 , term in x 4 (d) 1 x + 1 2 2 (1 – x) , term in x 2
x
126
02 STPM Math T T1.indd 126 3/28/18 4:21 PM