Number Sequences
Explore arithmetic sequences and geometric patterns, and learn how to find the rule and predict the next numbers.
What is a Sequence?
A sequence is a list of numbers that follow a rule or pattern. Each number in the list is called a term.
Counting in Steps
Rule: Add 3 each time. The next term is 18.
Arithmetic Sequences
In an arithmetic sequence, you add (or subtract) the same number each time. This constant difference is called the common difference.
Going Up (+7)
5, 12, 19, 26, 33, ...
Common difference = +7
Going Down (−4)
30, 26, 22, 18, 14, ...
Common difference = −4
Tip: To find the common difference, subtract any term from the next term. For example, 12 − 5 = 7.
Geometric Patterns (Doubling & Halving)
In a geometric pattern, each term is found by multiplying (or dividing) by the same number.
Doubling Pattern (×2)
Each term is multiplied by 2. The next term: 32 × 2 = 64.
Finding the Rule
To find the rule of a sequence, look at the difference between each pair of terms. If the differences are the same, it is arithmetic.
Example: 4, 9, 14, 19, 24
Rule: Start at 4, add 5 each time. Next term: 29.
Key Vocabulary
Sequence
An ordered list of numbers that follow a rule.
Term
Each number in a sequence (e.g. the 3rd term).
Common Difference
The constant amount added or subtracted between terms in an arithmetic sequence.
Rule
The instruction that tells you how to get the next term (e.g. "add 5").
Worked Examples
Find the next two terms: 7, 13, 19, 25, ...
Step 1: Find the common difference: 13 − 7 = 6
Step 2: Next term: 25 + 6 = 31
Step 3: Next term: 31 + 6 = 37
Answer: 31, 37
Find the next term: 3, 6, 12, 24, ...
Step 1: Check: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. Multiplied by 2 each time.
Step 2: Next term: 24 × 2 = 48
Answer: 48
What is the 10th term if the sequence starts at 2 and adds 5 each time?
Step 1: First few terms: 2, 7, 12, 17, 22, ...
Step 2: From the 1st term to the 10th, we add 5 a total of 9 times.
Step 3: 9 × 5 = 45. Then 2 + 45 = 47
Answer: The 10th term is 47.
Knowledge Check
Select the correct answer for each question.
Question 1
What is the next term in the sequence: 5, 11, 17, 23, ...?
Question 2
What is the common difference in the sequence: 50, 43, 36, 29, 22?
Question 3
What is the next term: 2, 6, 18, 54, ...?
Question 4
A sequence starts at 10 and adds 8 each time. What is the 6th term?
Question 5
Which sequence uses the rule "subtract 9"?
Key Concepts Summary
- ●A sequence is an ordered list of numbers following a rule.
- ●An arithmetic sequence adds or subtracts the same amount each time (the common difference).
- ●A geometric pattern multiplies or divides by the same number each time.
- ●To find the rule, look at the differences (or ratios) between consecutive terms.
- ●You can use the rule to predict future terms or find any term in the sequence.