Patterns And Sequences — 3–5 Math | Scholars & Parents

What Is This?

A sequence is an ordered list of numbers or shapes that follows a rule. A pattern is any repeating or predictable arrangement. Identifying the rule in a sequence lets you predict any term — including terms far down the list without writing out every number in between.

Why Does It Matter?

Pattern recognition is one of the most fundamental skills in mathematics. Sequences appear in multiplication tables, prime numbers, Fibonacci numbers, compound interest, computer algorithms, and natural phenomena. Understanding patterns also builds the foundation for algebra: finding a rule is exactly what writing an equation means.

Key Vocabulary

Term	Meaning
Sequence	An ordered list of numbers following a rule
Term	Each number in a sequence
Rule	The operation applied to get from one term to the next
Arithmetic sequence	Each term is found by adding or subtracting a fixed number
Geometric sequence	Each term is found by multiplying or dividing by a fixed number
Common difference	The fixed amount added or subtracted in an arithmetic sequence
Common ratio	The fixed amount multiplied or divided in a geometric sequence

Part 1 — Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms.

Increasing arithmetic sequences (add)

Example: 3, 7, 11, 15, 19, …

Rule: +4 each time (common difference = 4)
Next term: 19 + 4 = 23

Example: 5, 10, 15, 20, 25, …

Rule: +5 (the 5 times table)
10th term: 5 + (9 × 5) = 50

Decreasing arithmetic sequences (subtract)

Example: 100, 93, 86, 79, 72, …

Rule: −7 each time
Next term: 72 − 7 = 65

Finding a specific term

For an arithmetic sequence starting at a with common difference d:

nth term = a + (n − 1) × d

Example: Sequence 4, 7, 10, 13, … What is the 10th term?

a = 4, d = 3
10th term = 4 + (10 − 1) × 3 = 4 + 27 = 31 ✓

Part 2 — Geometric Sequences

A geometric sequence has a constant ratio — each term is multiplied (or divided) by the same number.

Multiplying sequences

Example: 2, 6, 18, 54, 162, …

Rule: ×3 each time (common ratio = 3)
Next term: 162 × 3 = 486

Example: 1, 2, 4, 8, 16, 32, …

Rule: ×2 (powers of 2)
8th term: 2⁷ = 128

Dividing sequences

Example: 256, 128, 64, 32, 16, …

Rule: ÷2 each time
Next term: 16 ÷ 2 = 8

Part 3 — Two-Operation and Other Patterns

Some sequences use more than one operation or a more complex rule.

Alternating patterns

Example: 1, 3, 2, 4, 3, 5, 4, 6, …

Odd positions: 1, 2, 3, 4 (add 1 to every other term)
Even positions: 3, 4, 5, 6 (add 1 to every other term)

Square numbers

1, 4, 9, 16, 25, 36, 49, … (n² for n = 1, 2, 3, …)

Triangular numbers

1, 3, 6, 10, 15, 21, … (add 1 more each time: +1, +2, +3, +4, +5, …)

Fibonacci-style sequences

Each term is the sum of the two terms before it.

1, 1, 2, 3, 5, 8, 13, 21, … (classic Fibonacci)
2, 5, 7, 12, 19, 31, … (same rule, different start)

Part 4 — Finding the Rule

To identify the rule in a sequence:

Step 1: Look at the differences between consecutive terms.

Step 2: If the differences are constant → arithmetic (add/subtract).

Step 3: If the ratios are constant → geometric (multiply/divide).

Step 4: If neither, look for a two-step rule or a special pattern.

Example: Find the rule for 3, 6, 12, 24, 48

Differences: 3, 6, 12, 24 — not constant
Ratios: 6÷3=2, 12÷6=2, 24÷12=2 — constant!
Rule: ×2 (geometric, ratio = 2)

Example: Find the rule for 2, 5, 10, 17, 26

Differences: 3, 5, 7, 9 — increasing by 2 each time
This is a second-difference pattern (quadratic): n² + 1

Part 5 — Shape and Visual Patterns

Sequences also appear in visual patterns. Count the shapes or tiles at each stage and look for the rule.

Example: A pattern of squares: Stage 1 = 1, Stage 2 = 4, Stage 3 = 9, Stage 4 = 16

Rule: n² (square numbers)
Stage 5: 5² = 25

Example: Matchstick pattern: 3, 5, 7, 9, …

Each new square adds 2 matchsticks
Rule: +2 (arithmetic, d = 2)
Stage 10: 3 + (10 − 1) × 2 = 3 + 18 = 21

Worked Examples

Worked Example 1: Find the next two terms

Sequence: 8, 13, 18, 23, 28, __, __

Differences: all +5 → arithmetic
Next terms: 28 + 5 = 33, 33 + 5 = 38 ✓

Worked Example 2: Find the missing term

Sequence: 3, __, 12, 24, 48

Ratios: 24÷12=2, 48÷24=2 → geometric, ×2
Missing term: 3 × 2 = 6 (then 6×2=12 ✓)

Worked Example 3: Find the 8th term

Sequence: 5, 9, 13, 17, … (d = 4, a = 5)

8th term = 5 + (8−1) × 4 = 5 + 28 = 33 ✓

Worked Example 4: Write the rule

Sequence: 80, 40, 20, 10, …

Each term is divided by 2 → rule: ÷2 (geometric, ratio = ½)
Next term: 10 ÷ 2 = 5 ✓

Worked Example 5 (Word Problem):

A training plan has a runner do 2 km on Day 1. Each day she runs 1.5 km more than the day before. How far does she run on Day 7?

a = 2, d = 1.5
Day 7 = 2 + (7−1) × 1.5 = 2 + 9 = 11 km ✓

Common Mistakes

Mistake 1: Assuming add when it's multiply 1, 2, 4, 8 — the difference increases (1, 2, 4) so it is not arithmetic. Check ratios: 2/1=2, 4/2=2, 8/4=2 → geometric ×2.

Mistake 2: Off-by-one in the nth term formula For the 5th term of 3, 7, 11, 15: it's 3 + (5−1)×4 = 3+16 = 19, not 3 + 5×4 = 23. The formula is a + (n−1)×d, not a + n×d.

Mistake 3: Not checking the rule against every term Always verify your rule works for at least three consecutive terms, not just the first two.

Mistake 4: Extending without checking the pattern type Extending 2, 4, 8 by adding 2 would give 10 — but the pattern is ×2, giving 16.

Key Takeaways

Arithmetic sequences: constant difference (add or subtract the same amount each time)
Geometric sequences: constant ratio (multiply or divide by the same amount each time)
To find a specific term: use nth term = a + (n−1) × d for arithmetic sequences
Always check your rule against multiple consecutive terms before extending
Square numbers (n²), triangular numbers, and Fibonacci sequences are common special patterns

Practice and Resources

Ready to practise? Try our Grade 3-5 Patterns and Sequences Worksheet covering arithmetic, geometric, and visual patterns. Test yourself with the Grade 3-5 Math Practice Test.