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What Is the Number Sequences?(With Examples)

A sequence is an ordered list of numbers following a specific pattern or rule. Each number is called a term. In an arithmetic sequence, the difference between consecutive terms is constant. In a geometric sequence, the ratio between consecutive terms is constant.

Grades 7-9Key Stage 3-4Årskurs 7-9Klasse 7-9

📖Definition

📐Formula

Arithmetic: aₙ = a₁ + (n-1)d; Geometric: aₙ = a₁ × rⁿ⁻¹

For arithmetic sequences, d is the common difference. For geometric sequences, r is the common ratio. aₙ is the nth term and a₁ is the first term.

📝Step-by-Step Guide

Identify the Pattern

Look for addition/subtraction (arithmetic) or multiplication/division (geometric) between terms.

Find the Common Difference or Ratio

Arithmetic: d = any term minus the previous term. Geometric: r = any term divided by the previous term.

Write the Formula

Use the appropriate formula for the sequence type.

Arithmetic: aₙ = a₁ + (n-1)d

Find Any Term

Substitute the term number (n) into the formula to find that term.

⚠️Common Mistakes to Avoid

Confusing arithmetic and geometric sequences
Using the wrong formula
Off-by-one errors with term numbers
Miscalculating the common difference or ratio
Forgetting n-1 in the formulas

✏️Practice Problems

Easy

Find the next term: 2, 5, 8, 11, ...

Answer: 14 (arithmetic, d = 3)

Medium

Find the 10th term of: 3, 6, 12, 24, ...

Answer: 1536 (geometric, r = 2)

Hard

In an arithmetic sequence, the 3rd term is 11 and the 7th term is 27. Find the first term.

Answer: a₁ = 3

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Worked Examples

Number Sequence Examplemedium

Number Patterns Linear Functions Exponents Series

Curriculum Alignment

CommonCore (HSF-BF.A.2)KS3KMK

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