CHAPTER 8: SERIES AND SEQUENCES
Chapter objectives:
• use the Binomial Theorem for expansion of (𝑎 + 𝑏)• for positive
integer 𝑛
𝑛
• use the general term x y 𝑎•^Æ 𝑏Æ , 0 ≤ 𝑟 ≤ 𝑛
𝑟
• recognise arithmetic and geometric progressions
• use the formulae for the 𝑛th term and for the sum of the first 𝑛 terms
to solve problems involving arithmetic or geometric progressions
• use the condition for the convergence of a geometric progression, and
the formula for the sum to infinity of a convergent geometric
progression
The binomial theorem for expansion
The binomial theorem for expansion states that:
𝑛 𝑛 𝑛 𝑛
(𝑎 + 𝑏)• = x y 𝑎• 𝑏k + x y 𝑎•^/𝑏/ + x y 𝑎•^0𝑏0 … + x y 𝑎k 𝑏•
0 1 2 𝑛
Where 𝑛 is a positive integer.
𝒏 𝒏!
x y= (pronounced n choose r)
𝒓 𝒓! (𝒏 − 𝒓)!
𝒏! = 𝒏 × (𝒏 − 𝟏) × (𝒏 − 𝟐) × … × 𝟑 × 𝟐 × 𝟏 (pronounced n factorial)
For example:
5! = 5 × 4 × 3 × 2 × 1 = 120
Similarly:
148
,21! = 21 × 20 × 19 × … × 3 × 2 × 1
NB: 𝑛! = 𝑛 × (𝑛 − 1)! = 𝑛 × (𝑛 − 1) × (𝑛 − 2)! etc.
Example 8.1
Find the first 4 terms in the expansion of (3 + 𝑥 0 )“ in ascending powers of
𝑥.
SOLUTION
Applying the binomial theorem of expansion:
6 6 6 6
(3 + 𝑥 0 )“ = x y 3“ (𝑥 0 )k + x y 3n (𝑥 0 )/ + x y 3m (𝑥 0 )0 + x y 3] (𝑥 0 )]
0 1 2 3
+⋯
(3 + 𝑥 0 )“ = 1 × 729 × 1 + 6 × 243 × 𝑥 0 + 15 × 81 × 𝑥 m + 20 × 27 × 𝑥 “
+⋯
∴ (3 + 𝑥 0)“ = 729 + 1458𝑥 0 + 1215𝑥 m + 540𝑥 “ + ⋯
Exercise 8.1
1. Find the first four terms of the expansion:
a. (1 + 𝑥)“ in ascending powers of 𝑥.
b. (2 − 𝑥)ª in ascending powers of 𝑥.
T //
c. x1 + y in ascending powers of 𝑥.
0
d. (1 − 𝑥 0 )¦ in ascending powers of 𝑥.
/ “
e. x𝑥 − y in descending powers of 𝑥.
T
149
, / n
f. x2𝑥 0 + y in descending powers of 𝑥.
0T
g. (1 + 𝑥 )(2 − 3𝑥 )“ in ascending powers of 𝑥.
T m
h. (3 − 2𝑥) x1 + y in ascending powers of 𝑥.
]
/ “
i. (1 + 𝑥 ) x𝑥 − y in descending powers of 𝑥.
T
/ / n
j. x 𝑥 − 3y x2𝑥 0 + y in descending powers of 𝑥.
0 0T
2. Find the coefficient of 𝑎 𝑏 in the expansion of (𝑎 + 2𝑏) “.
] ]
3. What is the value of 𝑎 given that in the expansion of (1 + 𝑎𝑥 )// the
coefficient of 𝑥 ] is 6 times the coefficient of 𝑥 0 .
4. Find the value of 𝑎 given that the coefficient of 𝑥 0 in the expansion of
(1 − 𝑎𝑥 )(2 + 𝑥 )“ is −32.
5. Find the term independent of 𝑥 in the expansion of the following
expression:
𝑥0 1 m
»2 + ¼ ‡𝑥 − ˆ
2 𝑥
The (𝒓 + 𝟏)th term
The (𝑟 + 1)th term in the expansion of (𝑎 + 𝑏)• is given by:
𝑛
𝑇ÆU/ = x y 𝑎•^Æ 𝑏Æ
𝑟
Where 0 < 𝑟 < 𝑛 and 𝑟 is an integer.
The equation for 𝑇ÆU/ can also be used to find the term in a certain power of
𝑥.
Example 8.2
Find the 7𝑡ℎ term in the expansion of (2 + 𝑥)/k
150
Chapter objectives:
• use the Binomial Theorem for expansion of (𝑎 + 𝑏)• for positive
integer 𝑛
𝑛
• use the general term x y 𝑎•^Æ 𝑏Æ , 0 ≤ 𝑟 ≤ 𝑛
𝑟
• recognise arithmetic and geometric progressions
• use the formulae for the 𝑛th term and for the sum of the first 𝑛 terms
to solve problems involving arithmetic or geometric progressions
• use the condition for the convergence of a geometric progression, and
the formula for the sum to infinity of a convergent geometric
progression
The binomial theorem for expansion
The binomial theorem for expansion states that:
𝑛 𝑛 𝑛 𝑛
(𝑎 + 𝑏)• = x y 𝑎• 𝑏k + x y 𝑎•^/𝑏/ + x y 𝑎•^0𝑏0 … + x y 𝑎k 𝑏•
0 1 2 𝑛
Where 𝑛 is a positive integer.
𝒏 𝒏!
x y= (pronounced n choose r)
𝒓 𝒓! (𝒏 − 𝒓)!
𝒏! = 𝒏 × (𝒏 − 𝟏) × (𝒏 − 𝟐) × … × 𝟑 × 𝟐 × 𝟏 (pronounced n factorial)
For example:
5! = 5 × 4 × 3 × 2 × 1 = 120
Similarly:
148
,21! = 21 × 20 × 19 × … × 3 × 2 × 1
NB: 𝑛! = 𝑛 × (𝑛 − 1)! = 𝑛 × (𝑛 − 1) × (𝑛 − 2)! etc.
Example 8.1
Find the first 4 terms in the expansion of (3 + 𝑥 0 )“ in ascending powers of
𝑥.
SOLUTION
Applying the binomial theorem of expansion:
6 6 6 6
(3 + 𝑥 0 )“ = x y 3“ (𝑥 0 )k + x y 3n (𝑥 0 )/ + x y 3m (𝑥 0 )0 + x y 3] (𝑥 0 )]
0 1 2 3
+⋯
(3 + 𝑥 0 )“ = 1 × 729 × 1 + 6 × 243 × 𝑥 0 + 15 × 81 × 𝑥 m + 20 × 27 × 𝑥 “
+⋯
∴ (3 + 𝑥 0)“ = 729 + 1458𝑥 0 + 1215𝑥 m + 540𝑥 “ + ⋯
Exercise 8.1
1. Find the first four terms of the expansion:
a. (1 + 𝑥)“ in ascending powers of 𝑥.
b. (2 − 𝑥)ª in ascending powers of 𝑥.
T //
c. x1 + y in ascending powers of 𝑥.
0
d. (1 − 𝑥 0 )¦ in ascending powers of 𝑥.
/ “
e. x𝑥 − y in descending powers of 𝑥.
T
149
, / n
f. x2𝑥 0 + y in descending powers of 𝑥.
0T
g. (1 + 𝑥 )(2 − 3𝑥 )“ in ascending powers of 𝑥.
T m
h. (3 − 2𝑥) x1 + y in ascending powers of 𝑥.
]
/ “
i. (1 + 𝑥 ) x𝑥 − y in descending powers of 𝑥.
T
/ / n
j. x 𝑥 − 3y x2𝑥 0 + y in descending powers of 𝑥.
0 0T
2. Find the coefficient of 𝑎 𝑏 in the expansion of (𝑎 + 2𝑏) “.
] ]
3. What is the value of 𝑎 given that in the expansion of (1 + 𝑎𝑥 )// the
coefficient of 𝑥 ] is 6 times the coefficient of 𝑥 0 .
4. Find the value of 𝑎 given that the coefficient of 𝑥 0 in the expansion of
(1 − 𝑎𝑥 )(2 + 𝑥 )“ is −32.
5. Find the term independent of 𝑥 in the expansion of the following
expression:
𝑥0 1 m
»2 + ¼ ‡𝑥 − ˆ
2 𝑥
The (𝒓 + 𝟏)th term
The (𝑟 + 1)th term in the expansion of (𝑎 + 𝑏)• is given by:
𝑛
𝑇ÆU/ = x y 𝑎•^Æ 𝑏Æ
𝑟
Where 0 < 𝑟 < 𝑛 and 𝑟 is an integer.
The equation for 𝑇ÆU/ can also be used to find the term in a certain power of
𝑥.
Example 8.2
Find the 7𝑡ℎ term in the expansion of (2 + 𝑥)/k
150
