1. Sequences
Objectives:
Grade 7: Write the terms of a sequence given the nth term
Grade 8: Find the nth term of a sequence or a series of diagrams

2. Sequences
Definitions:
Sequence
A list of numbers or diagrams that are connected in some way.
Term
A term is a number and / or variable(s) connected with
x and / or ÷ separated from anther term by an ‘+’ or ‘-’ operation.
nth term
The ‘general’ term used to describe a sequence, e.g. 3n + 1
If you are given the nth term you can find the terms of a
sequence.
Coefficient
The number preceding a letter – the number that is used to
multiply a letter. e.g. 2n the coefficient is 2

3. The term to term rule is the rule that connects numbers in a sequence:
Sequences
The term to term rule is the rule that connects numbers in a sequence:
e.g. Find the term to term rule for this sequence:
The dots show that the sequence continues
5, 7, 9, 11, ...
Each term (consecutive number in the sequence) is 2 more than the
one before it, so the rule is: +2

4. Sequences
Now try these:
Find the term to term rule for these sequences:
1) 3, 7, 11, 15, ...
2) 0, 5, 10, 15, ...
3) 16, 13, 10, 7, ...
4) 3, 1, -1, - 3, ...
5) 1, 2, 4, 8, 16,
6) 2, 3.5, 5, 6.5, ...
7) 0.01, 0.1, 1, 10, ... +4 +5 -3 -2
x 2
+1.5
x 10

5. Sequences
To write the terms of a sequence given the nth term
Given the expression: 2n + 3 write the first 5 terms
In this expression the letter n represents the term number and thus
if we substitute the term number for the letter n we will find value
that particular term.
The first 5 terms of the sequence will be using values for n of
1, 2, 3, 4 and 5
term 1
term 2
term 3
term 4
term 5
2 x 1 + 3
2 x 2 + 3
2 x 3 + 3
2 x 4 + 3
2 x 5 + 3 5 7 9 11 13

6. Sequences
Now try these:
Write the first 5 terms of these sequences:
1) n + 2
2) 2n + 5
3) 3n - 2
4) 5n + 3
5) -4n + 10
6) n2 + 2
7) 3n2
3, 4, 5, 6, 7
7, 9, 11, 13, 15
1, 4, 7, 10, 13
8, 13, 18, 23, 28
6, 2, - 2, - 6, - 10
3, 6, 11, 18, 27
3, 12, 27, 48, 75

7. n +2 and the sequence it generates:
Sequences
Look at the nth term
n +2 and the sequence it generates:
3, 4, 5, 6, 7
Find the term to term rule: +1
Look at the nth term
2n +5 and the sequence it generates:
7, 9, 11, 13, 15
Find the term to term rule: +2
You will notice that the coefficient in each expression is the
term to term rule:
n +2 rule +1
2n +5 rule +2
The coefficient of n is 1 because if we multiply n by 1 it is still n

8. Sequences
To find the nth term of a sequence or a series of diagrams
this shows us that the coefficient for a sequence is the term to
term rule, so we always find this first.
Example:
Find the nth term for the following sequence:
3, 7, 11, 15
The term to term rule is: +4
Therefore the coefficient of n is 4 so we write 4n
However, if we find the first 4 terms of this we get:
4, 8, 12, 16
This doesn’t give the correct sequence, but you will notice that
If you subtract 1 from each of these terms you get the correct sequence
The full nth term is therefore:
4n - 1

9. Sequences
To summarise finding the nth term:
Find the term to term rule
Find the term to term rule
Find how much you need to add or subtract
to get the correct sequence
Find how much you need to add or subtract
to get the correct sequence
Check your nth term works for the 2nd and 3rd terms
Try this:
Find the nth term for the following sequence: 5, 8, 11, 14 +3
Write: 3n
use n = 1 for the 1st term
the 1st term would be 3 x 1 = 3
We need to + 2 to make the correct
first term of 5
The nth term becomes 3n + 2
Check this in terms 2 and 3
3 x = 8
3 x = 11

10. Sequences
Now try these:
Find the nth term for these sequences:
1) 4, 6, 8, 10, ...
2) 1, 6, 11, 16, ...
3) 3, 10, 17, 24, ...
4) 10, 19, 28, 37, ...
5) 13, 10, 7, 4, ...
20, 14, 8, 2, ...
2n + 2
5n - 4
7n - 4
9n + 1
-3n + 16
-6n + 26

11. Sequences
Find the nth term of a series of diagrams
Here is a series of diagrams 5 9 13 17
Write the number of matches in each pattern
Now we have a sequence of numbers
from which we can find the nth term
The term to term rule is +4
The nth term is therefore
4n + 1

12. Sequences
Now try these:

13. Sequences
Now try these: 4 7 13 10
3n + 1 6 10 14 18
4n + 2 16 28 40
12n + 4