SEQUENCES AND SERIES
Unit 2 General Maths
DESCRIBING SEQUENCES
This topic investigates different types of patterns and how they can be manipulated
mathematically.
The dictionary describes a sequence as, ‘a number of things, actions, or events
arranged or happening in a specific order or having a specific connection’.
In maths, the term sequence is used to represent an ordered set of elements.
In this topic we will examine the relationships and patterns of these sets of data.
RECOGNISING PATTERNS
For these examples:
eg1)
2, 4, 6, 8……..
Describe the pattern in words;
Describe the pattern in mathematical terms;
State the next 3 numbers in the pattern.
Increasing by 2
+2
10, 12, 14
Doubling each number
x2
80, 160, 320
eg2)
5, 10, 20, 40…..
eg3)
1000, 500, 250…… Halving each number
÷2
125, 62.5, 31.25
USING A RULE TO GENERATE A
NUMBER PATTERN
For these examples:
eg1)
Use the following rules to write down the
first five numbers of each number pattern
Start with a 72 and divide by 2 each time.
72, 36, 18, 9, 4.5
eg2)
Start with 2, Multiply by 4 and subtract 3.
2, 5, 17, 65, 257
NOW DO
Chapter 5
Exercise 5A Questions 1 – 3
ARITHMETIC SEQUENCES
An ARITHMETIC SEQUENCE is one in which the difference between any two
consecutive terms is the same.
Are these Arithmetic Sequences?
eg. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Yes – the difference between consecutive terms is the same
eg. 0, 3, 6, 9, 12, 15, 18, 21, 24
Yes – the difference between consecutive terms is the same
eg. 2, 4, 5, 10, 11, 16
No – the difference between consecutive terms is NOT the same
eg. 2, 4, 8, 16, 32, 64
No – the difference between consecutive terms is NOT the same
ARITHMETIC SEQUENCES
The terms, in order, can be labelled
𝑡1 , 𝑡2 , 𝑡3 , … … 𝑡𝑛
Label the terms in the following examples:
eg. 2, 3, 4, 5, 6, 7, 8, 9, 10
𝑡1 = 2,
𝑡2 = 3,
𝑡3 = 4
𝑒𝑡𝑐 …
eg. 0, 3, 6, 9, 12, 15, 18, 21, 24
𝑡1 = 0,
𝑡2 = 3,
𝑡3 = 6
𝑒𝑡𝑐 …
eg.
𝑡1 = 4,
𝑡2 = 6,
𝑡3 = 8
𝑒𝑡𝑐 …
4, 6, 8, 10, 12, 14
ARITHMETIC SEQUENCES
We can label the first term in the sequence
′𝑎′
We can label the ‘common difference’ between consecutive terms
Label the arithmetic sequences with their ‘a’ and ‘d’ values
𝑎=2
eg. 2, 3, 4, 5, 6, 7, 8, 9, 10
eg. 0, 3, 6, 9, 12, 15, 18, 21, 24
eg.
4, 6, 8, 10, 12, 14
𝑑=1
𝑎=0
𝑎=4
𝑑=3
𝑑=2
′𝑑′
ARITHMETIC SEQUENCES
GENERATED BY RECURSION
A sequence can be generated by the repeated use of an instruction. This is known as
recursion. We can form equations to model recursion:
The current term given by 𝑛 can be represented by 𝑡𝑛
The following term is 𝑡𝑛+1 & the term before it is 𝑡𝑛−1
Using this, we can form an equation to model the recursion. The recursion relation is:
𝑡𝑛+1 = 𝑡𝑛 + 𝑑 ,
The next term
The current term
𝑡1 = 𝑎
The common
difference
The first term in
the sequence
Recursion relation
The next term
𝑡𝑛+1 = 𝑡𝑛 + 𝑑,
The current term
𝑡1 = 𝑎
The common
difference
Form recursion relations for the following:
eg. 2, 3, 4, 5, 6, 7, 8, 9, 10
𝑡𝑛+1 = 𝑡𝑛 + 1,
𝑡1 = 2
eg. 0, 3, 6, 9, 12, 15, 18, 21, 24
𝑡𝑛+1 = 𝑡𝑛 + 3,
𝑡1 = 0
eg.
𝑡𝑛+1 = 𝑡𝑛 + 0.5,
𝑡1 = 4
𝑡𝑛+1 = 𝑡𝑛 − 3,
𝑡1 = 30
4, 4.5, 5, 5.5, 6, 6.5
eg. 30, 27, 24, 21, 18, 15
The first term in
the sequence
NOW DO
Chapter 5
Recursion Worksheet – Exercise 1
FINDING TERMS OF AN
ARITHMETIC SEQUENCE
If we know the first term 𝑎, and the common difference 𝑑, we can find any number
within the arithmetic sequence.
The rule for finding a term in an arithmetic sequence is:
𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑
The value of the nth
term (the term we
are trying to find)
The first term in
the sequence
The common
difference
The position of
the term we are
trying to find in
the sequence
Rule for Arithmetic
Sequences
The value of the n
th
term (the term we
are trying to find)
eg. Consider the arithmetic sequence
𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑
The first term in
the sequence
The common
difference
The position of
the term we are
trying to find in
the sequence
22, 28, 34, 40, …..
a) Write a rule for the arithmetic sequence
𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑
𝑡𝑛 = 22 + 𝑛 − 1 6
b) What number would be at the 30th position?
𝑡𝑛 = 22 + 𝑛 − 1 6
𝑡30 = 22 + 30 − 1 6
𝑡30 = 22 + 29 6 = 22 + 174 = 196
Rule for Arithmetic
Sequences
The value of the n
th
term (the term we
are trying to find)
eg. Consider the arithmetic sequence
𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑
The first term in
the sequence
The common
difference
The position of
the term we are
trying to find in
the sequence
10, 13, 16, 19, 22, 25,…
a) Write a rule for the arithmetic sequence
𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑
𝑡𝑛 = 10 + 𝑛 − 1 × 3
b) What number would be at the 16th position?
𝑡𝑛 = 10 + 𝑛 − 1 × 3
𝑡16 = 10 + 16 − 1 × 3
𝑡16 = 10 + 15 × 3 = 10 + 45 = 55
Rule for Arithmetic
Sequences
The value of the n
th
term (the term we
are trying to find)
eg. Consider the arithmetic sequence
𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑
The first term in
the sequence
The common
difference
The position of
the term we are
trying to find in
the sequence
41, 37, 33, 29, 25,….
a) Write a rule for the arithmetic sequence
𝑡𝑛 = 𝑎 + 𝑛 − 1 𝑑
𝑡𝑛 = 41 + 𝑛 − 1 × −4
b) What number would be at the 11th position?
𝑡𝑛 = 41 + 𝑛 − 1 × −4
𝑡30 = 41 + 11 − 1 × −4
𝑡30 = 41 + 10 × −4 = 41 − 40 = 1
NOW DO
Chapter 5
Exercise 5B
Questions 1, 4a, 4b, 4d 4f, 5a, 5b, 5d, 5f
ARITHMETIC SERIES
An arithmetic series is the term used for the sum of all of the terms in an arithmetic
sequence. We can find the arithmetic series using one of two different methods.
If we know the first term 𝑎, and the last term 𝑙, the sum of the 𝑛 terms is given by:
The number of
terms in the
sequence
𝑛
𝑆𝑛 = ( 𝑎 + 𝑙 )
2
The first term in
the sequence
The last term in
the sequence
ARITHMETIC SERIES
An arithmetic series is the term used for the sum of all of the terms in an arithmetic
sequence. We can find the arithmetic series using one of two different methods.
If we know the first term 𝑎, the common difference 𝑑, the sum of the 𝑛 terms is given by:
The number of
terms in the
sequence
𝑛
𝑆𝑛 = ( 2𝑎 + 𝑛 − 1 𝑑)
2
The first term in
the sequence
The common
difference
The number of
terms in the
sequence
eg. A sequence starts at 4 and ends
with 32. If there are 8 terms in the
sequence, find the arithmetic series
(the sum) for the sequence.
𝑆𝑛 =
𝑛
2
=
8
2
𝑛
𝑆𝑛 = ( 2𝑎 + 𝑛 − 1 𝑑
2
The first term in
the sequence
The first term in
the sequence
(𝑎+𝑙)
4 + 32
= 4( 36)
= 144
The common
difference
The number of
terms in the
sequence
𝑛
𝑆𝑛 = ( 𝑎 + 𝑙 )
2
The last term in
the sequence
The number of
terms in the
sequence
eg. Find the sum of the first 20
values in the sequence 3, 5, 7, 9,…
𝑆𝑛 =
=
20
2
𝑛
2
𝑛
𝑆𝑛 = ( 2𝑎 + 𝑛 − 1 𝑑
2
The first term in
the sequence
The common
difference
( 2𝑎 + 𝑛 − 1 𝑑
The first term in
the sequence
2 × 3 + 20 − 1 2
= 10(6 + 19 × 2 )
= 10 6 + 38
= 10 × 44 = 440
The number of
terms in the
sequence
𝑛
𝑆𝑛 = ( 𝑎 + 𝑙 )
2
The last term in
the sequence
The number of
terms in the
sequence
eg. Find the sum of the first 18
values in the sequence 6, 11, 16, 21…
𝑆𝑛 =
=
18
2
𝑛
2
𝑛
𝑆𝑛 = ( 2𝑎 + 𝑛 − 1 𝑑
2
The first term in
the sequence
The common
difference
( 2𝑎 + 𝑛 − 1 𝑑
The first term in
the sequence
2 × 6 + 18 − 1 5
= 9(12 + 17 × 5 )
= 9 12 + 85
= 9 × 97 = 873
The number of
terms in the
sequence
𝑛
𝑆𝑛 = ( 𝑎 + 𝑙 )
2
The last term in
the sequence
The number of
terms in the
sequence
eg. Find the sum of the first 10 values
in the sequence 40, 36, 32, 28,…
𝑆𝑛 =
=
10
2
𝑛
2
𝑛
𝑆𝑛 = ( 2𝑎 + 𝑛 − 1 𝑑
2
The first term in
the sequence
The common
difference
( 2𝑎 + 𝑛 − 1 𝑑
The first term in
the sequence
2 × 40 + 10 − 1 × −4
= 5(80 + 9 × −4 )
= 5 80 − 36
= 5 × 44 = 220
The number of
terms in the
sequence
𝑛
𝑆𝑛 = ( 𝑎 + 𝑙 )
2
The last term in
the sequence
NOW DO
Chapter 5
Exercise 5C
Questions 1, 4, 6, 7, 8, 9a
ARITHMETIC SERIES
We can also use the arithmetic series to find the value of a term at any position 𝑛 within
the sequence.
Value of
the term n
𝑡𝑛 = 𝑆𝑛 − 𝑆𝑛−1
Arithmetic Series
until the term n
Arithmetic
series until term
before n
Value of
the term n
𝑡𝑛 = 𝑆𝑛 − 𝑆𝑛−1
Arithmetic Series
until the term n
Arithmetic
series until term
before n
eg. Find the value of the term at position 20 of the sequence 14, 17, 20, 23,….
𝑡𝑛 = 𝑆𝑛 − 𝑆𝑛−1
𝑡20 = 𝑆20 − 𝑆19
= 850 − 779
= 71
20
2
𝑆20 =
28 + 20 − 1 3
= 10(28 + 19 × 3
= 10 28 + 57
= 10 × 85
= 850
19
2
𝑆19 =
28 + 19 − 1 3
= 9.5(28 + 18 × 3
= 9.5 (28 + 54)
= 9.5(82)
= 779
NOW DO
Chapter 5
Exercise 5B
Questions 1, 4, 6, 7, 8, 9a, 11, 12a