KS3 Mathematics
A4 Sequences
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Contents
A4 Sequences
A4.1 Introducing sequences
A4.2 Describing and continuing sequences
A4.3 Generating sequences
A4.4 Finding the nth term
A4.5 Sequences from practical contexts
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Sequences from geometrical patterns
We can show many well-known sequences using geometrical
patterns of counters.
Even Numbers
2
4
6
8
10
5
7
9
Odd Numbers
1
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3
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Sequences from geometrical patterns
Multiples of Three
3
6
9
12
15
15
20
25
Multiples of Five
5
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10
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Sequences from geometrical patterns
Square Numbers
1
4
9
16
25
6
10
15
Triangular Numbers
1
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3
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Sequences with geometrical patterns
How could we arrange counters to represent the
sequence 2, 6, 12, 20, 30, . . .?
The numbers in this sequence can be written as:
1 × 2,
2 × 3,
3 × 4,
4 × 5,
5 × 6, . . .
We can show this sequence using a sequence of rectangles:
1 × 2 = 2 2 × 3 = 6 3 × 4 = 12 4 × 5 = 20
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5 × 6 = 30
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Powers of two
We can show powers of two like this:
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
Each term in this sequence is double the term before it.
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Powers of three
We can show powers of three like this:
31 = 3
32 = 9
33 = 27
34 = 81
35 = 243
36 = 729
Each term in this sequence is three times the term before it.
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Sequences that increase in equal steps
We can describe sequences by finding a rule that tells us
how the sequence continues.
To work out a rule it is often helpful to find the difference
between consecutive terms.
For example, look at the difference between each term in
this sequence:
3,
7,
+4
11,
+4
15
+4
19,
+4
23,
+4
27,
+4
31, . . .
+4
This sequence starts with 3 and increases by 4 each time.
Every term in this sequence is one less than a multiple of 4.
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Sequences that decrease in equal steps
Can you work out the next three terms in this sequence?
22,
–6
16,
10,
–6
–2,
4,
–6
–6
–8, –14, –20, . . .
–6
–6
–6
How did you work these out?
This sequence starts with 22 and decreases by 6 each time.
Each term in the sequence is two less than a multiple of 6.
Sequences that increase or decrease in equal steps
are called linear or arithmetic sequences.
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Sequences that increase in increasing steps
Some sequences increase or decrease in unequal steps.
For example, look at the differences between terms in this
sequence:
2,
6,
+1
8,
+2
11,
+3
15,
+4
20,
+5
26,
+6
33, . . .
+7
This sequence starts with 5 and increases by 1, 2, 3, 4, …
The differences between the terms form a linear sequence.
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Sequences that decrease in decreasing steps
Can you work out the next three terms in this sequence?
7,
6.9,
–0.1
6.7,
–0.2
6.4,
–0.3
6,
–0.4
5.5,
–0.5
4.9, 4.2, . . .
–0.6
–0.7
How did you work these out?
This sequence starts with 7 and decreases by 0.1, 0.2, 0.3,
0.4, 0.5, …
With sequences of this type it is often helpful to find a second
row of differences.
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Using a second row of differences
Can you work out the next three terms in this sequence?
1,
3,
+2
8,
+5
+3
16,
+8
+3
27,
+11
+3
41,
+14
+3
58,
+17
+3
78, . . .
+20
+3
Look at the differences between terms.
A sequence is formed by the differences so we look at the
second row of differences.
This shows that the differences increase by 3 each time.
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Sequences that increase by multiplying
Some sequences increase or decrease by multiplying or
dividing each term by a constant factor.
For example, look at this sequence:
2,
4,
×2
8,
×2
16,
×2
32,
×2
64,
×2
128, 256, . . .
×2
×2
This sequence starts with 2 and increases by multiplying the
previous term by 2.
All of the terms in this sequence are powers of 2.
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Sequences that decrease by dividing
Can you work out the next three terms in this sequence?
512, 256,
÷4
÷4
64,
16,
÷4
4,
÷4
1,
÷4
0.25, 0.125, . . .
÷4
÷4
How did you work these out?
This sequence starts with 512 and decreases by dividing
by 4 each time.
We could also continue this sequence by multiplying by
each time.
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1
4
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Fibonacci-type sequences
Can you work out the next three terms in this sequence?
1,
1,
2,
3,
5,
8,
13,
21,
34,
1+1
1+2
3+5
5+8
8+13
13+21
21+13
55, . . .
21+34
How did you work these out?
This sequence starts 1, 1 and each term is found by
adding together the two previous terms.
This sequence is called the Fibonacci sequence after the
Italian mathematician who first wrote about it.
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Describing and continuing sequences
Here are some of the types of sequence you may come
across:
Sequences that increase or decrease in equal steps.
These are called linear or arithmetic sequences.
Sequences that increase or decrease in unequal steps
by multiplying or dividing by a constant factor.
Sequences that increase or decrease in unequal steps
by adding or subtracting increasing or decreasing numbers.
Sequences that increase or decrease by adding together
the two previous terms.
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Continuing sequences
A number sequence starts as follows
1, 2, . . .
How many ways can you think of continuing the sequence?
Give the next three terms and the rule for each one.
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Finding missing terms
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Name that sequence!
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