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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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Introducing sequences
In maths, we call a list of numbers in order a sequence.
Each number in a sequence is called a term.
4, 8, 12, 16, 20, 24, 28, 32, . . .
1st term
6th term
If terms are next to each other they are referred to as
consecutive terms.
When we write out sequences, consecutive terms are
usually separated by commas.
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Infinite and finite sequences
A sequence can be infinite. That means it continues forever.
For example, the sequence of multiples of 10,
10, 20 ,30, 40, 50, 60, 70, 80, 90 . . .
is infinite. We show this by adding three dots at the end.
If a sequence has a fixed number of terms it is called a
finite sequence.
For example, the sequence of two-digit square numbers
16, 25 ,36, 49, 64, 81
is finite.
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Sequences and rules
Some sequences follow a simple rule that is easy to describe.
For example, this sequence
2, 5, 8, 11, 14, 17, 20, 23, 26, 29, …
continues by adding 3 each time. Each number in this
sequence is one less than a multiple of three.
Other sequences are completely random.
For example, the sequence of winning raffle tickets in a
prize draw.
In maths we are mainly concerned with sequences of
numbers that follow a rule.
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Naming sequences
Here are the names of some sequences which you may
know already:
2, 4, 6, 8, 10, . . .
Even Numbers (or multiples of 2)
1, 3, 5, 7, 9, . . .
Odd numbers
3, 6, 9, 12, 15, . . .
Multiples of 3
5, 10, 15, 20, 25 . . .
Multiples of 5
1, 4, 9, 16, 25, . . .
Square numbers
1, 3, 6, 10,15, . . .
Triangular numbers
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Ascending sequences
When each term in a sequence is bigger than the one
before the sequence is called an ascending sequence.
For example,
The terms in this ascending sequence increase in equal
steps by adding 5 each time.
2,
7,
12, 17, 22, 27, 32, 37, . . .
+5
+5
+5
+5
+5
+5
+5
The terms in this ascending sequence increase in unequal
steps by starting at 0.1 and doubling each time.
0.1, 0.2, 0.4, 0.8, 1.6, 3.2, 6.4, 12.8, . . .
×2
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×2
×2
×2
×2
×2
×2
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Descending sequences
When each term in a sequence is smaller than the one
before the sequence is called a descending sequence.
For example,
The terms in this descending sequence decrease in equal
steps by starting at 24 and subtracting 7 each time.
24, 17, 10,
3, –4, –11, –18, –25, . . .
–7
–7
–7
–7
–7
–7
–7
The terms in this descending sequence decrease in
unequal steps by starting at 100 and subtracting 1, 2, 3, …
100, 99, 97, 94, 90, 85, 79, 72, . . .
–1
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–2
–3
–4
–5
–6
–7
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Sequences from real-life
Number sequences are all
around us.
Some sequences, like the ones
we have looked at today follow
a simple rule.
Some sequences follow more
complex rules, for example, the
time the sun sets each day.
Some sequences are completely random, like the sequence of
numbers drawn in the lottery.
What other number sequences can be made from real-life
situations?
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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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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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Sequence grid
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Generating sequences from flow charts
A sequence can be given by a flow chart. For example,
START
Write down 3.
Add on 1.5.
Write down the answer.
No
Is the answer
more than 10?
This flow chart
generates the
sequence
3, 4.5, 6, 6.5, 9.
This sequence has
only five terms.
It is finite.
Yes
STOP
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Generating sequences from flow charts
START
Write down 5.
Subtract 2.1.
Write down the answer.
No
This flow chart
generates the
sequence
5, 2.9, 0.8, –1.3, –3.4.
Is the answer
less than -5?
Yes
STOP
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Generating sequences from flow charts
START
Write down 200.
Divide by 2.
Write down the answer.
No
This flow chart
generates the
sequence
200, 100, 50, 25,
12.5, 6.25.
Is the answer
less than 4?
Yes
STOP
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Generating sequences from flow charts
START
Write down 3 and 4.
Add together the two
previous numbers.
Write down the answer.
No
This flow chart
generates the
sequence
3, 4, 7, 11, 18, 29,
47, 76.
Is the answer
more than 100?
Yes
STOP
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Predicting terms in a sequence
Usually, we can predict how a sequence will continue by
looking for patterns.
For example,
87, 84, 81, 78, . . .
We can predict that this sequence continues by subtracting
3 each time.
However, sequences do not always continue as we would
expect.
For example,
A sequence starts with the numbers 1, 2, 4, . . .
How could this sequence continue?
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Continuing sequences
Here are some different ways in which the sequence might
continue:
1
2
+1
1
4
+2
2
×2
7
+3
4
×2
11
+4
8
×2
16
+5
16
×2
22
+6
32
×2
64
×2
We can never be certain how a sequence will continue
unless we are given a rule or we can justify a rule from a
practical context.
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Continuing sequences
This sequence continues by adding 3 each time.
1
4
+3
7
+3
10
+3
13
+3
16
+3
19
+3
We can say that rule for getting from one term to the next
term is add 3.
This is called the term-to-term rule.
The term-to-term rule for this sequence is +3.
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Using a term-to-term rule
Does the rule +3 always produce the same sequence?
No, it depends on the starting number.
For example, if we start with 2 and add on 3 each time we
have,
2,
5,
8,
11,
14,
17,
20,
23, . . .
If we start with 0.4 and add on 3 each time we have,
0.4, 3.4, 6.4, 9.4, 12.4, 15.4, 18.4, 21.4, . . .
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Writing sequences from term-to-term-rules
A term-to-term rule gives a rule for finding each term of
a sequence from the previous term or terms.
To generate a sequence from a term-to-term rule we must
also be given the first number in the sequence.
For example,
1st term
Term-to-term rule
5
Add consecutive even numbers starting with 2.
This gives us the sequence,
5
7
+2
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11
+4
17
+6
27
+10
39
+12
53 . . .
+14
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Sequences from a term-to-term rule
Write the first five terms of each sequence
given the first term and the term-to-term rule.
1st term
10
Add 3
10, 13, 16, 19, 21
100
Subtract 5
100, 95, 90, 85, 80
3
Double
3, 6, 12, 24, 48
5
Multiply by 10
5, 50, 500, 5000, 50000
7
Subtract 2
7, 5, 3, 1, –1
Add 0.1
0.8, 0.9, 1.0, 1.1, 1.2
0.8
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Term-to-term rule
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Sequences from position-to-term rules
Sometimes sequences are arranged in a table like this:
Position
1st
2nd
3rd
4th
5th
6th
…
nth
Term
3
6
9
12
15
18
…
3n
We can say that each term can be found by multiplying
the position of the term by 3.
This is called a position-to-term rule.
For this sequence we can say that the nth term is 3n,
where n is a term’s position in the sequence.
What is the 100th term in this sequence? 3 × 100 = 300
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Sequences from position-to-term rules
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Writing sequences from position-to-term rules
The position-to-term rule for a sequence is very useful
because it allows us to work out any term in the sequence
without having to work out any other terms.
We can use algebraic shorthand to do this.
We call the first term T(1), for Term number 1,
we call the second term T(2),
we call the third term T(3), . . .
we call the nth term T(n).
T(n) is called the the nth term or the general term.
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Writing sequences from position-to-term rules
For example, suppose the nth term of a sequence is 4n + 1.
We can write this rule as:
T(n) = 4n + 1
Find the first 5 terms.
T(1) = 4 × 1 + 1 = 5
T(2) = 4 × 2 + 1 = 9
T(3) = 4 × 3 + 1 = 13
T(4) = 4 × 4 + 1 = 17
T(5) = 4 × 5 + 1 = 21
The first 5 terms in the sequence are: 5, 9, 13, 17 and 21.
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Writing sequences from position-to-term rules
If the nth term of a sequence is 2n2 + 3.
We can write this rule as:
T(n) = 2n2 + 3
Find the first 4 terms.
T(1) = 2 × 12 + 3 = 5
T(2) = 2 × 22 + 3 = 11
T(3) = 2 × 32 + 3 = 21
T(4) = 2 × 42 + 3 = 35
The first 4 terms in the sequence are: 5, 11, 21, and 35.
This sequence is a quadratic sequence.
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Sequence generator – linear sequences
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Sequence generator – non-linear sequences
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Sequences and rules
Which rule is best?
The term-to-term rule?
The position-to-term rule?
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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 of multiples
All sequences of multiples can be generated by adding the
same amount each time. They are linear sequences.
For example, the sequence of multiples of 5:
5,
10,
+5
15,
+5
20,
+5
25,
+5
30
+5
35
+5
40 …
+5
can be found by adding 5 each time.
Compare the terms in the sequence of multiples of 5 to
their position in the sequence:
Position
1
×5
Term
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5
2
×5
10
3
4
×5
15
×5
20
5
…
×5
25
n
×5
…
5n
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Sequences of multiples
The sequence of multiples of 3:
3,
6,
+3
9,
+3
12,
+3
15,
+3
18,
+3
21,
+3
24, …
+3
can be found by adding 3 each time.
Compare the terms in the sequence of multiples of 3 to
their position in the sequence:
Position
1
×3
Term
3
2
3
×3
6
4
×3
9
×3
12
5
…
×3
15
n
×3
…
3n
The nth term of a sequence of multiples is always dn,
where d is the difference between consecutive terms.
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Sequences of multiples
The nth term of a sequence of multiples is always dn,
where d is the difference between consecutive terms.
For example,
The nth term of 4, 8, 12, 16, 20, 24 … is 4n
The 10th term of this sequence is 4 × 10 = 40
The 25th term of this sequence is 4 × 25 = 100
The 47th term of this sequence is 4 × 47 = 188
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Finding the nth term of a linear sequence
The terms in this sequence
4,
7,
+3
10,
+3
13,
+3
16,
+3
19,
+3
22,
+3
25 …
+3
can be found by adding 3 each time.
Compare the terms in the sequence to the multiples of 3.
Position
1
2
×3
Multiples
of 3
3
Term
4
3
×3
6
+1
×3
9
+1
7
4
5
×3
12
+1
10
+1
13
×3
15
+1
16
…
n
×3
3n
+1
… 3n + 1
Each term is one more than a multiple of 3.
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Finding the nth term of a linear sequence
The terms in this sequence
1,
6,
+5
11,
+5
16,
+5
21,
+5
26,
+5
31,
+5
36 …
+5
can be found by adding 5 each time.
Compare the terms in the sequence to the multiples of 5.
Position
1
2
×5
Multiples
of 5
5
Term
1
3
×5
10
–4
–4
6
4
×5
15
–4
11
5
×5
20
–4
16
×5
25
–4
21
…
n
×5
5n
–4
… 5n – 4
Each term is four less than a multiple of 5.
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Finding the nth term of a linear sequence
The terms in this sequence
5,
3,
–2
–1,
1,
–2
–2
–3,
–2
–5,
–2
–7,
–2
–9 …
–2
can be found by subtracting 2 each time.
Compare the terms in the sequence to the multiples of –2.
Position
1
2
× –2
Multiples
of –2
–2
Term
5
3
× –2
–4
+7
× –2
–6
+7
3
4
+7
1
5
× –2
–8
+7
–1
…
× –2
× –2
–2n
–10
+7
–3
n
+7
…
7 – 2n
Each term is seven more than a multiple of –2.
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Arithmetic sequences
Sequences that increase (or decrease) in equal steps are
called linear or arithmetic sequences.
The difference between any two consecutive terms in an
arithmetic sequence is a constant number.
When we describe arithmetic sequences we call the
difference between consecutive terms, d.
We call the first term in an arithmetic sequence, a.
For example, if an arithmetic sequence has a = 5 and d = -2,
We have the sequence:
5,
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3,
1,
-1,
-3,
-5,
...
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The nth term of an arithmetic sequence
The rule for the nth term of any arithmetic sequence is of
the form:
T(n) = an + b
a and b can be any number, including fractions and
negative numbers.
For example,
T(n) = 2n + 1
Generates odd numbers starting at 3.
T(n) = 2n + 4
Generates even numbers starting at 6.
T(n) = 2n – 4
Generates even numbers starting at –2.
T(n) = 3n + 6
Generates multiples of 3 starting at 9.
T(n) = 4 – n
Generates descending integers starting at 3.
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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 practical contexts
The following sequence of patterns is made from L-shaped tiles:
Number of
Tiles
4
8
12
16
The number of tiles in each pattern form a sequence.
How many tiles will be needed for the next pattern?
We add on four tiles each time. This is a term-to-term rule.
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Sequences from practical contexts
A possible justification of this rule is that each shape has four
‘arms’ each increasing by one tile in the next arrangement.
The pattern give us multiples of 4:
1 lot of 4
2 lots of 4
3 lots of 4
4 lots of 4
The nth term is 4 × n or 4n.
Justification: This follows because the 10th term would be
10 lots of 4.
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Sequences from practical contexts
Now, look at this pattern of blocks:
Number of
Blocks
4
7
10
13
How many blocks will there be in the next shape?
We add on 3 blocks each time.
This is the term-to term rule.
Justification: The shapes have three ‘arms’ each increasing
by one block each time.
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Sequences from practical contexts
How many blocks will there be in the 100th arrangement?
We need a rule for the nth term.
Look at pattern again:
1st pattern
2nd pattern
3rd pattern
4th pattern
The nth pattern has 3n + 1 blocks in it.
Justification: The patterns have 3 ‘arms’ each increasing by
one block each time. So the nth pattern has 3n blocks in the
arms, plus one more in the centre.
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Sequences from practical contexts
So, how many blocks will there be in the 100th pattern?
Number of blocks in the nth pattern = 3n + 1
When n is 100,
Number of blocks = (3 × 100) + 1 = 301
How many blocks will there be in:
a) Pattern 10? (3 × 10) + 1 = 31
b) Pattern 25? (3 × 25) + 1 = 76
c) Pattern 52? (3 × 52) + 1 = 156
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Paving slabs 1
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Paving slabs 2
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Paving slabs 2
The number of blue tiles form the sequence 8, 13, 18, 32, . . .
Pattern
number
Number of
blue tiles
1
2
3
8
13
18
The rule for the nth term of this sequence is
T(n) = 5n + 3
Justification: Each time we add another yellow tile we
add 5 blue tiles. The +3 comes from the 3 tiles at the
start of each pattern.
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Dotty pattern 1
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Dotty pattern 2
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Leapfrog investigation
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