Patterns & Sequences
Pupil Notes and worked examples
Key Skill 1: Find particular terms in a sequence
Key Skill 2: Find missing terms in a sequence
Key Skill 3: Determine the rule for a sequence
A sequence is a list of things, usually numbers, that follows a rule.
If you can find the rule then you can continue a sequence.
e.g.
1, 3, 5, 7, …
The rule of this sequence is + 2
2, 7, 10, 13, 16, 19, …
The rule of this sequence is + 3
100, 98, 96, 94, 92, …
The rule of this sequence is – 2
1, 3, 9, 27, …
The rule of this sequence is x 3
If you continue the sequence you can find specific terms of the sequence.
e.g.
rule + 2
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, …
rule + 3
2, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, …
rule - 2
100, 98, 96, 94, 92, 90, 88, 96, 94, 92, 90, 88, 86, 84, …
rule x 3
1, 3, 9, 27, 81, 243, 729, 2187, …
Key Skill 4: Given the rule for a sequence, and its first term, list the set of numbers generated
Rule
First term
Sequence
rule + 2
1
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, …
rule + 3
2
2, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, …
rule - 2
100
rule x 3
1
100, 98, 96, 94, 92, 90, 88, 96, 94, 92, 90, 88, 86, …
1, 3, 9, 27, 81, 243, 729, 2187, …
Key Skill 5: Determine the formula for a sequence
Finding a formula is a useful way to find specific terms of a sequence without generating all the
previous terms of a sequence.
We shall list the position of each term of the sequence using the letter n
e.g. For the sequence 1, 3, 5, 7, 9, 11, …
n
sequence
1
2
3
4
5
6
1
3
5
7
9
11
1
2
3
4
5
6
1
3
5
7
9
11
The 6th term of the
sequence is 11
Find the rule for the sequence
n
sequence
+2
+2
+2
+2
+2
Multiply the top row by the rule
2xn
n
sequence
2
1
4
2
6
3
8
4
10
5
12
6
1
3
5
7
9
11
How do you go from 2 x n to the sequence?
Answer: take away one - 1
2xn
n
2
1
4
2
6
3
8
4
10
5
12
6
sequence
2xn–1
1
1
3
3
5
5
7
7
9
9
11
11
This produces the formula 2 x n – 1 for the sequence 1, 3, 5, 7, 9, 11, …
This formula can be simplified to 2n – 1