Sequences
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
or
× 0.5
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 4 and multiply by 2 each time.
4, 8, 16, 32, 64
GEOMETRIC SEQUENCE – changes
by a ‘common ratio’
eg2)
Start with 21 and add 6 each time.
21, 27, 33, 39, 45
ARITHMETIC SEQUENCE –
changes by a ‘common difference’
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
eg. 2, 3, 4, 5, 6, 7, 8, 9, 10
𝑎=2
eg. 0, 3, 6, 9, 12, 15, 18, 21, 24
eg.
4, 6, 8, 10, 12, 14
𝑎=0
𝑎=4
𝑑=1
𝑑=3
𝑑=2
′𝑑′
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 – common difference
eg. 0, 3, 6, 9, 12, 15, 18, 21, 24
Yes – common difference
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 at each position 𝑡𝑛 , 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
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:
The common
difference
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
The value of the nth
term (the term we are
trying to find)
The first term in
the sequence
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)
eg1. Consider the arithmetic sequence
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
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 + 𝑑 𝑛 − 1
𝑡𝑛 = 22 + 6 𝑛 − 1
b) What number would be at the 30th position?
𝑙𝑒𝑡 𝑛 = 30:
𝑡𝑛 = 22 + 6 𝑛 − 1
𝑡30 = 22 + 6 30 − 1
𝑡30 = 22 + 6 29 = 22 + 174 = 196
Rule for Arithmetic
Sequences
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
eg2. Consider the arithmetic sequence
10, 13, 16, 19, 22, 25,…
a) Write a rule for the arithmetic sequence
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
𝑡𝑛 = 10 + 3 𝑛 − 1
b) What number would be at the 16th position?
𝑡𝑛 = 10 + 3 𝑛 − 1
𝑙𝑒𝑡 𝑛 = 16:
𝑡16 = 10 + 3 16 − 1
𝑡16 = 10 + 3 15 = 10 + 45 = 55
Rule for Arithmetic
Sequences
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
eg3. Consider the arithmetic sequence
41, 37, 33, 29, 25,….
a) Write a rule for the arithmetic sequence
𝑡𝑛 = 41 − 4 𝑛 − 1
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
b) What number would be at the 11th position?
𝑙𝑒𝑡 𝑛 = 30:
𝑡𝑛 = 41 − 4 𝑛 − 1
𝑡30 = 41 − 4 11 − 1
𝑡30 = 41 − 4 10
= 41 − 40 = 1
Rule for Arithmetic
Sequences
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
eg4. Consider a sequence which starts at 14 and each number increases by 4.5.
Find the first 5 numbers in the sequence.
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
𝑡𝑛 = 14 + 4.5 𝑛 − 1
𝑙𝑒𝑡 𝑛 = 2:
𝑙𝑒𝑡 𝑛 = 3:
𝑡2 = 14 + 4.5 2 − 1
𝑡2 = 14 + 4.5 1
= 18.5
𝑡3 = 14 + 4.5 3 − 1
𝑡3 = 14 + 4.5 2
= 14 + 9
= 23
etc….
OR use your calculator
Rule for Arithmetic
Sequences
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
Eg4 (continued).
Consider a sequence which starts at 14 and each number increases by 4.5.
Find the first 5 numbers in the sequence.
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
𝑡𝑛 = 14 + 4.5 𝑛 − 1
OR use your calculator in
another way..
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
Eg4 (continued) Using the SEQUENCES function.
Choose Explicit
𝑡𝑛 = 14 + 4.5 𝑛 − 1
Change these values for ‘n’
to produce a table of values
as small or large as you
require
NOW DO
Exercise 6.2
Q1, 2, 3abc, 4abc, 5, 6, 7
FINDING OTHER VARIABLES IN AN
ARITHMETIC SEQUENCE
We can rearrange the rule
to find a, d or n:
𝑡𝑛 = 𝑎 + 𝑑 𝑛 − 1
Finding the first number in a sequence:
Finding the common difference:
Finding the position of the term:
𝑑=
𝑎 = 𝑡𝑛 − 𝑑 𝑛 − 1
𝑡𝑛 − 𝑎
𝑛−1
𝑛=
𝑡𝑛 − 𝑎
𝑑
+1
Finding other variables
Eg1. Find the common difference of the arithmetic sequence which has a 1st term of 20
and a 6th term of 75
Finding the common difference:
𝑑 =?
𝑛=6
𝑡6 = 75
a = 20
𝑑=
𝑑=
=
𝑡𝑛 − 𝑎
𝑛−1
75−20
6−1
55
5
= 11
Finding other variables
Eg2. A sequence has a first term of 8 and a common difference of 12.5. Which term has
the value of 108?
Finding the position of the term:
𝑛 =?
𝑎=8
𝑡𝑛 = 108
𝑑 = 12.5
=
100
+
12.5
=9
1
𝑛=
𝑡𝑛 − 𝑎
𝑑
𝑛=
108−8
+
12.5
+1
1
Finding other variables
Eg3. Find the first value of the arithmetic sequence which has a common difference of 7
and a 40th term of 408
Finding the first number in a sequence:
𝑎 =?
𝑛 = 40
𝑡40 = 408
𝑑=7
𝑎 = 𝑡𝑛 − 𝑑 𝑛 − 1
𝑎 = 408 − 7 40 − 1
= 408 − 7 39
= 135
NOW DO
Exercise 6.2
Q1, 2, 3abc, 4abc, 5, 6, 7, 8abc, 15ac, 16ac, 17a
Creating a table of values and
using it to plot the sequence
eg1. Consider the arithmetic sequence
𝑡𝑛 = 1 + 3 𝑛 − 1
a) Draw a table of values showing the term number and the term value for the first 5
terms in the sequence.
b) Plot the graph of the sequence
Creating a table of values and
using it to plot the sequence
eg2. Consider the arithmetic sequence
𝑡𝑛 = 10 + 5 𝑛 − 1
a) Draw a table of values showing the term number and the term value for the first 5 terms
in the sequence.
b) Plot the graph of the sequence
c) What is the value of the 21st
term in the sequence?
Creating a table of values and
using it to plot the sequence
eg2. Consider the arithmetic sequence
𝑡𝑛 = 10 + 5 𝑛 − 1
a) Draw a table of values showing the term number and the term value for the first 5 terms
in the sequence.
b) Plot the graph of the sequence
c) What is the value of the 21st
term in the sequence?
Creating a table of values and
using it to plot the sequence
eg3. Consider the arithmetic sequence
𝑡𝑛 = 15 − 2.5 𝑛 − 1
a) Draw a table of values showing the term number and the term value for the first 8
terms in the sequence.
Creating a table of values and
using it to plot the sequence
eg3. Consider the arithmetic sequence
𝑡𝑛 = 15 − 2.5 𝑛 − 1
a) Draw a table of values showing the term number and the term value for the first 8 terms
in the sequence.
b) Plot the graph of the sequence
c) What is the value of the 32nd
term in the sequence?
NOW DO
Exercise 6.2
Q9, 10, 19, 20