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Chapter 6: SEQUENCE AND PATTERNS
Learning objectives
By the end of this topic, the learners should be able to
• Recognise and generate number patterns
• Explain how to generate a sequence
• Use number machines to generate a sequence
• Describe a general rule when a pattern is given
• Determine terms in a sequence
We often need to spot a pattern in order to predict what will happen next. In maths, the correct name for a pattern of numbers is called a SEQUENCE. In this topic therefore you will learn how to identify and describe general rules for patterns. You will be able to determine a term in the sequence and nd the missing numbers in the sequence
6.1 Draw and Identify the Patterns
For any pattern it is important to try to spot what is happening before you can predict the next number.
Activity : Identifying the number patterns
(a) 3; 6; 9; 12; · · ·
To obtain the next number in the sequence, we add 3 to the previous number. The numbers in this sequence are multiples of 3.
(b) 7; 14; 21; 28; · · ·
To obtain the next number in the sequence, we add 7 to the previous number. The numbers in this sequence are multiples of 7.
(c) The table below shows the natural numbers from 1 to 100.
(a) The fth multiple of 11 is 55
(b) The nineth multiple of 11 is 99
(c) The fth multiple of 5 is 25
(d) The 12th multiple of 5 is 60
(e) The 20th multiple of 5 is 100
6.1 Exercise Set
1. Identify the pattern in the following sequences
(a) 3; 6; 9; 12; · · ·
(b) 20; 40; 60; 80; · · ·
(c) 1; 2; 3; 4; 5; · · ·
(d) 1; 4; 9; 16; · · ·
(e) 3; 7; 10; 17; 27; · · ·
(f) 1; 3; 6; 10; 15; · · ·
2. Write down the next 4 terms of each of these sequences:
(a) 4; 7; 10; 13; 16; 19; · · ·
(b) 5; 11; 17; 23; 29; 35; · · ·
(c) 6; 8; 11; 15; 20; 26; · · ·
(d) 10; 14; 20; 28; 38; · · ·
(e) 24; 23; 21; 18; 14; 9; · · ·
(f) 2; 12; 21; 29; 36; 42; · · ·
3. Using a number square box below ,answer the questions after the table.
(a) Identify the number whose multiples have been shaded yellow
(b) The 3rd multiple of nine is ......
(c) The 9th multiple of nine is ......
(d) The 20th multiple of nine is ......
(e) The 5th multiple of ...... is 60
(f) The 8th multiple of ... is 56.
(g) The 400th multiple of nine is ......
(h) State the multiples of 4 ,that are in the table
(i) The ......th multiple of nine is 1800
(j) The ......th multiple of nine is 2970.
5. (i) Write down the rst eight multiples of 8.
(ii) Write down the rst 8 multiples of 6.
(iii) What is the smallest number that is a multiple of both 6 and 8?
(iv) If 48 is the nth multiple of 12, what is n?
(v) If 96 is the nth multiple of 12, what is n ?
6. Three multiples of a number are 34, 170 and 255. What is the number?
7. Three multiples of a number are 38, 95 and 133. What is the number?
8. Four multiples of a number are 49, 77, 133 and 203. What is the number?
9. The number 24 is a multiple of 2 and a multiple of 3. What other numbers is it a multiple of?
10. Two multiples of a number have been shaded on this number square. What is the number?
11. 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
12. Color the numbers
6.2 Describing the General Rule
Activity : Finding the Next Term in the sequence
Find the next numbers in the sequences below
(a) 3; 6; 9; 12; · · ·
To obtain the next number in the sequence, we add 3 to the previous number.
(b) 7; 14; 21; 28; · · ·
To obtain the next number in the sequence, we add 7 to the previous number.
c) 24; 21; 18; 15; · · ·
To obtain the next number in the sequence, we subtract 3 from the previous number.
(d) 3; 9; 27; 81; · · ·
To obtain the next number in the sequence, we multiply 3 with the previous number.
6.2 Exercise Set
1. Find the next three numbers(terms) in each of the following sequences
(a) 1; 3; 5; 7; · · ·
(b) 2; 5; 8; 11; · · ·
(c) 4; 2; 0; -2; · · ·
(d) 1; 2; 6; 24; 120; · · ·
(e) 13; 23; 33; 43; · · ·
(f) 5; 9; 13; 17; · · ·
(g) 1; 2; 4; 8; · · ·
(h) 13; 1 6; 12 1 ; 24 1 ; · · ·
(i) 4; 9; 25; 49; · · ·
(j) 18; 30; 42; 54; 66; · · ·
(k) 1; -21; 1 4; -81; · · ·
(l) 4:1; 4:7; 5:3; 5:9; 6:5; · · ·
(m) 3:42; 3:56; 3:70; 3:84; · · ·
(n) 10; 9:5; 9; 8:5; 8; 7:5; · · ·
2. Fill in the missing numbers
(a) 6; 11; :::; 21; :::; · · ·
(b) :::; :::; 41; 36; 31; 26 · · ·
(c) 2; 4; :::; 16; 32; · · ·
(d) 100; 81; 64; :::; 36; · · ·
(e) -2; :::; -8; :::; -14; · · ·
(f) 0; 1:5; 4; :::; 12; · · ·
(g) 1; 7; 17; :::; 49; · · ·
6.3 Generating Number Sequence
Activity : Generating a sequence This involves using a formulae to generate sequences for given values.
EXAMPLES
What sequence do you generate by using the following formula?.Take n = 1; 2; 3; 4; 5; · · ·
1. 2n
we substitute the value of n ,in the formula given
Therefore the generated sequence is 2; 4; 6; 8; 10; · · ·
2. 8n - 5
we substitute the value of n ,in the formula given
Therefore the generated sequence is 3; 11; 19; 27; 35; · · ·
3. 6n + 2
we substitute the value of n ,in the formula given
Therefore the generated sequence is 8; 14; 20; 26; 32; · · ·
Input/Output machine
Some math problems contain a pattern, so once you nd a pattern then you can make a rule that will solve the problem for a given input. Therefore we put numbers into the machine[input], and the machine uses an operation (add, subtract, multiply or divide) to give us a result[output].
EXAMPLES
What number comes out of each of these number machines?
6.3 Exercise Set
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 7 terms of the sequence given by each of these formulae. Take n = 1; 2; 3; · · ·
(a) 4n - 1
(b) 7n - 1
(c) 8n
(d) 9n + 3
(e) 0.5n
4. Taking n = 1; 2; 3; · · · , what is
(a) the 10th term of the sequence 2n - 1?
(b) the 8th term of the sequence 3n ?
(c) the 5th term of the sequence 4n + 1?
(d) the 7th term of the sequence 5n + 2 ?
5. Draw a double(input/output) machine that yields the following outputs.[The formula(rule)
for each sequence must be written clearly]
(a) 1; 2; 3; 4; 5; · · ·
(b) 7; 14; 21; · · ·
(c) 2; 5; 8; 11; 14; · · ·
(d) 6; 11; 16; 21; 26; · · ·
(e) 4; 9; 14; 19; 24; · · ·
(f) 102; 202; 302; 402; 502; · · ·
6.4 Formulae for General Terms
It is very helpful not only to be able to write down or generate the next few terms in a sequence, but also to be able to write down any nth term .for example, the 100th term. Therefore this involves generating a formula using a given sequence.
Activity : Identifying the nth term
EXAMPLE
1. For the sequence3; 7; 11; 15; · · · Find
(a) the next three terms.
To obtain the next number(term) in the sequence, we add 4 to the previous number.
(b) the 100th term.
Therefore the 100th term is 399
(c) the 1000th term.
To obtain the 1000th term,we can base on the nth term.
The formula for a general term, i.e. the nth term.
2. For the sequence6; 11; 16; 21 · · · Find
(a) the next three terms.
To obtain the next number(term) in the sequence, we add 5 to the previous number.
(b) the 100th term.
Therefore the 100th term is 251
(c) the 1000th term.
To obtain the 1000th term, we can base on the nth term.
The formula for a general term, i.e. the nth term.
6.4 Exercise Set
1. Given the following sequences
(a) 0; 1; 2; 3 · · ·
(b) 1; 3; 5; 7 · · ·
(c) 5; 10; 15; 20 · · ·
(d) 2; 7; 12; 17 · · ·
(e) -2; -5; -8; -11 · · ·
Find
(i) the next three terms in the sequences
(ii) the 100th,20th and 31st terms for each of the sequence
(iii) the formula for the nth term of each of the sequence
2. What is the nth term of the sequence 2; 5; 10; 17; 26 · · · ?
3. Write down the rst 6 multiples of 8 and the formula for the nth term of the sequence .
4. What is the nth term of the sequence 1; 4; 9; 16 · · ·
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.
• 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.
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