1. Naming sequences Name these sequences: 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,
Multiples of 5
This can be done as an oral activity. It may also be useful for pupils to copy these number patterns into their books.
As each number sequence is revealed ask the name of the sequence before revealing it.
Year 7 pupils should have met square numbers and triangular numbers in Year 6. They will be revisited in more detail later in the year.
Pupils may describe the pattern of square numbers either as adding 3, adding 5, adding 7 etc. (i.e. adding consecutive odd numbers) or as 1 × 1, 2 × 2, 3 × 3, 4 × 4, 5 × 5 etc.
Pupils may need help verbalizing a rule to generate triangular numbers (i.e. add together consecutive whole numbers).
1, 4, 9, 16, 25, . . .
Square numbers
1, 3, 6, 10,15, . . .
Triangular numbers

2. Objective
By the end of the lesson you should be able to generate number sequences given a rule. Be able to use and generate position-to-term rules.

3. 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, , , , , , , 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.
Stress the difference between sequences that increase in equal steps (linear sequences) and sequences that increase in unequal steps (non-linear) sequences.
0.1, 0.2, , , , , , , . . . ×2 ×2 ×2 ×2 ×2 ×2 ×2

4. 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, , , –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, …
Stress the difference between sequences that decrease in equal steps (linear sequences) and sequences that decrease in unequal steps (non-linear) sequences.
100, 99, , , , , , 72, . . . –1 –2 –3 –4 –5 –6 –7

5. 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, , , , , , , . . . +4 +4 +4 +4 +4 +4 +4
Remind pupils that the sequence of multiples of 4 increases by adding 4 each time.
Ask pupils how this sequence is related to multiples of 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.

6. Sequences that decrease in equal steps
Can you work out the next three terms in this sequence?
22, 16, , , –2,
–8,
–14,
–20, –6 –6 –6 –6 –6 –6 –6
How did you work these out?
This sequence starts with 22 and decreases by 6 each time.
Introduce the word difference and encourage pupils to find the difference between consecutive terms.
Remind pupils that they must check that every number in the sequence obeys the same rule.
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.

7. Fibonacci-type sequences
Can you work out the next three terms in this sequence?
1, 1, 2, 3, 5, , ,
21,
34,
55,
1+1
1+2
3+5
5+8
8+13
13+21
21+13
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.
Sequences of this type must be generated by two numbers.
Ask pupils if this type of sequence can be descending. If the sequence is generated by two negative numbers then it will be a descending sequence.
The Fibonacci sequence appears in many situations in nature. Ask pupils to research some examples on the Internet.
This sequence is called the Fibonacci sequence after the Italian mathematician who first wrote about it.

8. 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.
Discuss each of these different type of sequence asking pupils to give examples for each one.
The second type of sequence is a sequence of powers or geometric sequence, the third type is a quadratic sequence, and the fourth is a Fibonacci-type sequence.
These can be broadly divided into two types: Sequences that increase or decrease in equal steps – linear sequences – and sequences that increase or decrease in unequal steps- non-linear sequences.
Ask pupils how we can use the differences between consecutive terms to help us to recognize each type.
In the first type of sequence the differences are constant.
In the second type of sequence the differences form another geometric sequence.
In the third type the differences form a linear sequence and so the second row of differences are constant.
In the fourth type the differences form the same sequence.
Sequences that increase or decrease by adding together
the two previous terms.

9. 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
Term-to-term rule 10
Add 3
10,
13,
16,
19, 21
100
Subtract 5
100,
95,
90,
85, 80 3
Double 3, 6,
12,
24, 48
Edit the numbers in this slide to produce more or less challenging examples. 5
Multiply by 10 5,
50,
500,
5000,
50000 7
Subtract 2 7, 5, 3, 1, –1
0.8
Add 0.1
0.8,
0.9,
1.0,
1.1,
1.2

10. 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), . . .
Explain the algebraic notation. Reassure pupils that using letters is a good way to save us lots of writing!
The T stands for term and the number in the bracket is the position of the term in the sequence. n can be any whole number.
Ask,
What do we call the 10th term of a sequence? (T(10))
What do we call the 450th term? And so on.
Explain again that we find the value of a term by substituting its position number into the rule for the nth term.
we call the nth term T(n).
T(n) is called the the nth term or the general term.

11. 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 × = 5
T(2) =
4 × = 9
T(3) =
4 × = 13
Repeat that T(1) is short way of writing the first term. To find the value of T(1) substitute 1 into the rule 4n + 1. To find the value of T(2) substitute 2 into the rule 4n + 1 etc.
What do you notice about this sequence? (It goes up 4 each time. An even better answer is: it is the numbers from the 4 times table with 1 added on each time.)
Ask pupils to use the rule to work out the value of the 10th term, the 50th term, the 243rd term, etc.
T(4) =
4 × = 17
T(5) =
4 × = 21
The first 5 terms in the sequence are: 5, 9, 13, 17 and 21.

12. 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 × = 5
T(2) =
2 × = 11
T(3) =
2 × = 21
Stress that in a linear sequence the n can only be raised to the power of 1 (though this is not usually written). In a quadratic sequence n can be raised to the power of 1 or the power of 2.
T(4) =
2 × = 35
The first 4 terms in the sequence are: 5, 11, 21, and 35.
This sequence is a quadratic sequence.

13. 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.
Stress that once we know a position-to-term rule we can find any term in the sequence given its position in the sequence.
Ask pupils to give other terms in the sequence with the position-to-term rule 3n.
For example, what is the 15th term in the sequence?
You could also ask pupils to give you the position of a given term in the sequence using inverse operations. For example,
42 is a term is this sequence. What position is it in?
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

14. 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, , , , , … +5 +5 +5 +5 +5 +5 +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:
Ask pupils, Can you find a rule linking the position of the term to the term?
Position 1 5 2 10 3 15 4 20 5 25 … n
× 5
× 5
× 5
× 5
× 5
× 5
Term 5n

15. Finding the nth term of a linear sequence
The terms in this sequence
4, , , , , , , … +3 +3 +3 +3 +3 +3 +3
can be found by adding 3 each time.
Compare the terms in the sequence to the multiples of 3.
Position 1 2 3 4 5 … n
× 3
× 3
× 3
× 3
× 3
× 3
Multiples of 3
Explain that this sequence increases by 3 each time like the 3 times table (multiples of 3, in other words). We therefore compare the position numbers to multiples of 3. 3 6 9 12 15 3n
+ 1
+ 1
+ 1
+ 1
+ 1
+ 1
Term 4 7 10 13 16 …
3n + 1
Each term is one more than a multiple of 3.

16. Finding the nth term of a linear sequence
The terms in this sequence
1, , , , , , , … +5 +5 +5 +5 +5 +5 +5
can be found by adding 5 each time.
Compare the terms in the sequence to the multiples of 5.
Position 1 2 3 4 5 … n
× 5
× 5
× 5
× 5
× 5
× 5
Multiples of 5
Explain that this sequence increases by 5 each time like the 5 times table (multiples of 5, in other words). We therefore compare the position numbers to multiples of 5. 5 10 15 20 25 5n
– 4
– 4
– 4
– 4
– 4
– 4
Term 1 6 11 16 21 …
5n – 4
Each term is four less than a multiple of 5.

17. Finding the nth term of a linear sequence
The terms in this sequence
5, , , –1, –3, –5, –7, –9 … –2 –2 –2 –2 –2 –2 –2
can be found by subtracting 2 each time.
Compare the terms in the sequence to the multiples of –2.
Position 1 2 3 4 5 … n
× –2
× –2
× –2
× –2
× –2
× –2
Multiples of –2
Explain that this sequence decreases by 2 each time. We therefore compare the position numbers to multiples of –2.
Explain that it is preferable not to have a negative sign at the beginning of an expression. We therefore write –2n + 7 as 7 – 2n. –2 –4 –6 –8
–10
–2n
+ 7
+ 7
+ 7
+ 7
+ 7
+ 7
Term 5 3 1 –1 –3 …
7 – 2n
Each term is seven more than a multiple of –2.

18. 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.
Explain what is meant by consecutive terms (terms that are next to each other in the sequence) and constant number (the same number each time).
Reinforce verbally that the difference between consecutive terms is always the same number because we are adding on the same amount each time.
We call the amount we add on each time, d, for difference.
Before revealing the third sentence, ask,
If we want to generate an arithmetic sequence what else do we need to know, apart from how much to add on each time?
Suppose we wanted to generate a sequence that went up 4 each time.
We need a start number – the first term in the sequence.
For example, if an arithmetic sequence has a = 5 and d = -2,
We have the sequence: 5, 3, 1,
-1,
-3,
-5,
. . .

19. 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.
For each example discuss the sequence generated by asking pupils to substitute values for n to give the first few terms of the sequence.
Assure pupils that T(n) = 4 – n has the form T(n) = an + b.
In this case, a is –1 and b is 4. T(n) = -n + 4 is equivalent to T(n) = 4 – n. We write it the other way round because it is ‘neater’ if there isn’t a negative sign at the beginning.
Remember: -1 n is just written as –n.
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.

20. 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
4n is the position-to-term rule or rule for the nth term.
It can also be called the general term of the sequence.
Remember n is the position of the term in the sequence. n can be any whole number bigger than 1.
Another way to say this rule, then, is 4 times the position number.
Whenever you give a rule for a pattern you should try to justify it to prove it works.
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.