Number Patterns and Sequences — How to Find the Rule

By MathPal Editorial Team·Published Apr 21, 2026

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

A sequence is an ordered list of numbers that follows a rule. In this guide you will learn to identify arithmetic and geometric sequences, find common differences and ratios, predict the next terms, and write formulas to find any term in the sequence without listing every one.

Theory

Arithmetic sequences

In an arithmetic sequence, each term is found by adding the same number to the previous term. That fixed number is called the common difference ($d$).

$2, \; 5, \; 8, \; 11, \; 14, \ldots \quad d = +3$

$20, \; 15, \; 10, \; 5, \; 0, \ldots \quad d = -5$

To find $d$: subtract any term from the term after it: $d = a_n - a_{n-1}$.

nth term formula for arithmetic sequences:

$a_n = a_1 + (n-1)d$

where $a_1$ is the first term and $n$ is the term position.

Geometric sequences

In a geometric sequence, each term is found by multiplying the previous term by the same number. That fixed number is called the common ratio ($r$).

$3, \; 6, \; 12, \; 24, \; 48, \ldots \quad r = 2$

$81, \; 27, \; 9, \; 3, \; 1, \ldots \quad r = \frac{1}{3}$

To find $r$: divide any term by the term before it: $r = \dfrac{a_n}{a_{n-1}}$.

nth term formula for geometric sequences:

$a_n = a_1 \times r^{n-1}$

Other patterns

Not all sequences are arithmetic or geometric. Some follow other rules:

• Square numbers: 1, 4, 9, 16, 25, … ($n^2$)
• Triangular numbers: 1, 3, 6, 10, 15, … (add 1, then 2, then 3, …)
• Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, … (each term = sum of the two before it)

Worked Examples

Example 1 — Identify and extend an arithmetic sequence

Sequence: 7, 12, 17, 22, …

Step 1: Check differences: $12-7=5$, $17-12=5$, $22-17=5$. Common difference $d = 5$.

Step 2: Next term: $22 + 5 = 27$.

Step 3: nth term formula: $a_n = 7 + (n-1) \times 5 = 7 + 5n - 5 = 5n + 2$.

Check: $a_1 = 5(1)+2 = 7$ ✓, $a_4 = 5(4)+2 = 22$ ✓.

Example 2 — Find the 15th term

Arithmetic sequence starting at 4 with common difference 6.

$a_{15} = 4 + (15-1) \times 6 = 4 + 84 = 88$

Example 3 — Identify a geometric sequence and find the next term

Sequence: 5, 15, 45, 135, …

Step 1: Check ratios: $\frac{15}{5}=3$, $\frac{45}{15}=3$, $\frac{135}{45}=3$. Common ratio $r = 3$.

Step 2: Next term: $135 \times 3 = 405$.

Step 3: nth term: $a_n = 5 \times 3^{n-1}$.

Check: $a_4 = 5 \times 3^3 = 5 \times 27 = 135$ ✓.

Example 4 — Find the missing term

Sequence: 6, ___, 24, 48, …

Step 1: Check if geometric: $\frac{48}{24} = 2$, so $r = 2$.

Step 2: Work backwards: $24 \div 2 = 12$. Also check: $6 \times 2 = 12$ ✓.

Answer: The missing term is 12.

Common Mistakes

Mistake 1 — Confusing arithmetic and geometric

❌ Sequence 3, 6, 12, 24: assuming $d = 3$ (arithmetic).

✅ Check by subtraction first: differences are 3, 6, 12 — not equal, so it's NOT arithmetic. Check ratios: $\frac{6}{3}=2$, $\frac{12}{6}=2$. It's geometric with $r=2$.

Mistake 2 — Applying the nth term formula from the wrong starting point

❌ Using $a_n = a_1 \times r^n$ instead of $a_n = a_1 \times r^{n-1}$.

✅ The exponent is $n-1$ because the first term ($n=1$) uses $r^0 = 1$, not $r^1$.

Mistake 3 — Forgetting that $d$ can be negative or fractional

❌ Assuming all arithmetic sequences increase.

✅ A decreasing sequence like 100, 93, 86, 79 is arithmetic with $d = -7$.

Practice Problems

Problem 1: Find the next two terms: 3, 8, 13, 18, …

Show Answer

$d = 5$. Next terms: 23, 28.

Problem 2: Find the 20th term of the arithmetic sequence 2, 5, 8, 11, …

Show Answer

$d = 3$, $a_1 = 2$.

$a_{20} = 2 + (20-1) \times 3 = 2 + 57 = \mathbf{59}$

Problem 3: Identify the type and find the next two terms: 256, 64, 16, 4, …

Show Answer

Geometric, $r = \frac{1}{4}$.

Next terms: $4 \times \frac{1}{4} = 1$, then $1 \times \frac{1}{4} = \frac{1}{4}$.

1, 1/4

Problem 4: Find the 8th term of the geometric sequence 2, 6, 18, …

Show Answer

$r = 3$, $a_1 = 2$.

$a_8 = 2 \times 3^7 = 2 \times 2187 = \mathbf{4374}$

Problem 5: Describe the pattern: 1, 4, 9, 16, 25, …

Show Answer

These are perfect squares: $1^2, 2^2, 3^2, 4^2, 5^2, \ldots$

The next term is $6^2 = 36$.

Summary

• An arithmetic sequence adds a constant (common difference $d$) each time: $a_n = a_1 + (n-1)d$.
• A geometric sequence multiplies by a constant (common ratio $r$) each time: $a_n = a_1 \times r^{n-1}$.
• To identify the type: check differences for arithmetic; check ratios for geometric.
• Not all patterns fit these two types — look for square numbers, triangular numbers, or recursive rules.

Related Topics

• Exponents and Powers — needed for geometric sequence nth term formula
• Variables and Algebraic Expressions — writing general rules with variables
• Integers — Operations and Number Line — arithmetic sequences with negative differences

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