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Topic 6:
SEQUENCE AND PATTERNS
By the end of this topic, you should be able to:
i) draw and identify the patterns.
ii) describe a general rule of a given pattern.
iii) describe a sequence.
iv) determine a term in a sequence.
v) find the missing numbers in a given sequence.
Introduction
In this topic you will learn how to identify and describe general rules for patterns. You will be able to determine a term in the sequence and find the missing numbers in the sequence.
Sub-topic 6.1: Draw and Identify the Patterns
Activity 6.1: Identifying number patterns
In groups, work in pairs.
Look at the following sequences, how can you get the next number?
Compare your answers with other members.
i) 3, 6, 9, 12, 15, …
ii) 2, 4, 6, 8, 10, 12, …
In (i), in order to get the next number, you add 2 to the previous number. The numbers in this sequence are multiples of 3. Sequence (ii), represents the multiples of 2.
Exercise
State the multiples of 3 found in this table:
This square shows multiples of a number. What is this number?
Write down the numbers that should go in each of these boxes. The
number square will help you with some of them.
a) The fifth multiple of … is …
b) The …th multiple of … is 36
c) The 12th multiple of … is …
d) The 20th multiple of … is …
e) The …th multiple of … is 96.
f) The 100th multiple of … is …
Solution
a) the 5th multiple of 4 is 20
b) the 9th multiple of 4 is 36
c) the 12th multiple of 4 is 48
d) the 20th multiple of 4 is 80
e) the 24th multiple of 4 is 96
f) the 100th multiple of 4 is 400
Exercise
1. On a number square like this one, shade all the multiples of 6. Then
answer the questions after the table.
a) What is the 4th multiple of 6?
b) What is the 10th multiple of 6?
c) What is the 12th multiple of 6?
d) What is the 100th multiple of 6?
2. The multiples of a number have been shaded on this square. What
is the number?
Copy each statement about these multiples and write down the numbers that go in the spaces.
a) The 3rd multiple of … is …
b) The 9th multiple of … is …
c) The 200th multiple of … is …
d) The …th multiple of … is 66
e) The … th multiple of … is 330.
3.
a) Write down the first 8 multiples of 8.
b) Write down the first 8 multiples of 6.
c) What is the smallest number that is a multiple of both 6 and 8?
d) What are the next two numbers that are multiples of both 6 and 8?
4. a) Write down the first 6 multiples of 12.
b) What is the 10th multiple of 12?
c) What is the 100th multiple of 12?
d) What is the 500th multiple of 12?
e) If 48 is the nth multiple of 12, what is n?
f) If 96 is the nth multiple of 12, what is n ?
5. a) What multiples have been shaded in this number square?
b) What is the first multiple not shown in the number square?
6. a) Explain why 12 is a multiple of 6 and 4.
b) Is 12 a multiple of any other numbers?
7. The number 24 is a multiple of 2 and a multiple of 3. What other numbers is it a multiple of?
8. Two multiples of a number have been shaded on this number square. What is the number?
9. Two multiples of a number have been shaded on this number square
a) What is the number?
b) What is the 19th multiple of this number?
10. Three multiples of a number are 34, 170 and 255. What is the number?
11. Three multiples of a number are 38, 95 and 133. What is the number?
12. Four multiples of a number are 49, 77, 133 and 203. What is the number?
Sub-topic 6.2: Describing the General Rule
Activity 6.2: Finding the Next Term
In your, groups work in pairs.
Can you use the given numbers of the sequence to deduce the pattern and hence find the next term?
Example: What are the next 3 numbers in the sequence:
a) 12, 17, 22 …?
b) 50, 47, 44, 41, 38, …?
Compare your answers with other group members
Solution
a) To find the pattern, it is usually helpful to first find the differences between each term i.e. the difference between 12 and 17 is 5; the difference between 17 and 22 is 5.
So the next term is found by adding 5 to the previous term. This gives you 27, 32, 37.
b) Again you find the difference between:
i) 50 and 47 is -3.
ii) 47 and 44 is -3.
iii) 44 and 41 is -3.
iv) 41 and 38 is -3.
So, the next term is found by taking away 3 from the previous term, giving you 35, 32, 29.
Exercise
1. Copy the following exercise and find the sequence in each case,
giving the next three numbers.
a) 18, 30, 42, 54, 66, …
b) 4.1, 4.7, 5.3, 5.9, 6.5, …
c) 8, 14, 20,…, 32, …
d) 3, 11, …, 27, 35, …
e) 3.42, 3.56, 3.70, 3.84, 3.98, …
f) 10, 9.5, 9, 8.5, 8, 7.5, …
2. Copy each sequence and fill in the missing numbers.
a) 2, 4, …, 16, 32, …
b) 100, 81, 64, …, 36, …
c) 6, 9, …, 21, 30, 30, …
d) 0, 1.5, 4, …, 12, …
e) 1, 7, 17, …, 49, …
Sub-topic 6.3: Generating Number Sequence
Activity 6.3: Generating a sequence
In your groups work in pairs.
You can use formulae to generate sequences. For example, the formula 5n, with n = 1, 2, 3, 4, … generates the sequence 5x1, 5x2, 5x3, 5x4, …
The sequence generated is 5, 10, 15, 20, …
Example: What sequence do you generate by using the following formula?
a) 5n – 1
b) 6n + 2
Solution
a) putting n = 1, 2, 3, 4, … gives 4, 9, 14, 19, …
b) putting n = 1, 2, 3, 4, … gives 8, 14, 20, 26, …
You can find the formula for this sequence, 11, 21, 31, 41, 51, 61, …
How you can find the sequence. The sequence begins with 11, and 11 =
10 + 1. Continue to add 10 each time the formula is 10n + 1.
Compare your answers with other members in the group.
Exercise
1. What number comes out of each of these number machines?
2. The sequence 1, 2, 3, 4, 5, … is put into each number machine. What does each machine do?
3. Write down the first 5 terms of the sequence given by each of these formulae:
a) 9n b) 12n c) 2n + 4 d) 3n – 1 e) 3n - 2
4. a) What is the 10th term of the sequence 2n + 1?
b) What is the 8th term of the sequence 3n + 6?
c) What is the 5th term of the sequence 4n + 1?
d) What is the 7th term of the sequence 5n – 1 ?
5. Draw double machines that could be used to get each of these
sequences from 1, 2, 3, 4, 5 …
Also write down the formula for each sequence of the following:
a) 5, 9, 13, 17, 21, …
b) 2, 5, 8, 11, 14, …
c) 6, 11, 16, 21, 26, …
d) 4, 9, 14, 19, 24, …
e) 102, 202, 302, 402, 502, …
Sub-topic 6.4: Formulae for General Terms
Activity 6.4 : Identifying the nth term
In your groups work in pairs.
Note: It is very helpful not only to be able to write down the next few
terms in a sequence, but also to be able to write down, for example,
the 100th or even the 1000th term.
Example: For the sequence 3, 7, 11, 15, …, …
Find:
a) the next three terms.
b) the 100th term.
c) the 1000th term.
Answer
a) You can see that 4 is added each time to get the next
term.
So you obtain 19, 23, 27.
b) To find the 100th term, starting at 3, you add 3 to 4 times ninety
nine times giving
3 + 4 x 99 = 3 + 396 = 399
c) Similarly, the 1000th term is
3 + 4 x 999 = 3 + 3996 = 3999
I can go one step further and write down the formula for a general term, i.e. the nth term. This is 3 + 4 x (n – 1) = 3 + 4n - 4 = 4n – 1.
Compare your answers with other members of the group and the examples given.
Exercise
1. For each sequence, write down the difference between each term and formula for the nth term.
a) 3, 5, 7, 9, 11, …
b) 5, 11, 17, 23, 29, …
c) 4, 7, 10, 13, 16, …
d) 2, 5, 8, 11, 14, …
e) 6, 10, 14, 18, 22, …
2. a) Write down the first 6 multiples of 11.
b) What is the formula for the nth term of the sequence of the multiples of 11?
c) What is the formula for the nth term of this sequence?
3. The formula for the nth term of this sequence is n².
1, 4, 9, 16, 25, …
What is the formula for the nth term of the following sequences?
a) 0, 3, 8, 15, 24, …
b) 10, 13, 18, 25, 34,
c) 2, 8, 18, 32, 50, …
d) 1, 8, 27, 64, 125, …
Situation of Integration
There is a family in the neighbourhood of your school. The family has a
rectangular compound on which they want to put up a hedge around.
The hedge shall be made up of plants of different colours.
Support: Physical instruments like hoes, machetes, tape measure
Resources: Knowledge of construction of figures like rectangles,
patterns, sequences
Task: The family requests you to plant the hedge around their
rectangular compound so that it looks beautiful.
Explain how you will plant the hedge, making sure that the plants at
the corners of the compound are the same in terms of colour.
Discuss whether there are other ways of planting the hedge.
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