Number Sequence Calculator

Generate and analyze mathematical sequences including arithmetic, geometric, Fibonacci, prime numbers, and custom patterns. Perfect for mathematics, programming, and pattern analysis.

Sequence Settings

Sequence Type

Number of Terms

First Term (a₁)

Common Difference (d)

Formula:

aₙ = a₁ + (n-1) × d

Quick Presets

Sequence Results

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Ready to Generate

Select a sequence type and click generate to see the results.

How Number Sequences Work

Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms. The nth term is calculated using: aₙ = a₁ + (n-1) × d

Example: 2, 5, 8, 11, 14... (first term = 2, common difference = 3)

The sum of the first n terms is: Sₙ = n/2 × (2a₁ + (n-1)d)

Geometric Sequences

A geometric sequence has a constant ratio between consecutive terms. The nth term is: aₙ = a₁ × r^(n-1)

Example: 3, 6, 12, 24, 48... (first term = 3, common ratio = 2)

The sum of the first n terms is: Sₙ = a₁ × (1 - r^n) / (1 - r) for r ≠ 1

Types of Number Sequences

Special Sequences

Fibonacci:

Each term is the sum of the two preceding ones

Prime Numbers:

Natural numbers greater than 1 with exactly two factors

Perfect Squares:

Numbers that are squares of integers (1, 4, 9, 16...)

Polynomial Sequences

Triangular Numbers:

Sum of first n natural numbers: n(n+1)/2

Perfect Cubes:

Numbers that are cubes of integers (1, 8, 27, 64...)

Factorial:

Product of all positive integers up to n

Applications

Mathematics:

Pattern recognition, series analysis

Computer Science:

Algorithm analysis, data structures

Finance:

Compound interest, annuity calculations

Example Calculations

Arithmetic Sequence Example

Find the 10th term of the sequence: 3, 7, 11, 15...

First term (a₁) = 3, Common difference (d) = 4

a₁₀ = 3 + (10-1) × 4 = 3 + 36 = 39

Geometric Sequence Example

Find the 6th term of the sequence: 2, 6, 18, 54...

First term (a₁) = 2, Common ratio (r) = 3

a₆ = 2 × 3^(6-1) = 2 × 3⁵ = 2 × 243 = 486

Fibonacci Sequence Example

The classic Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...

Each term is the sum of the two preceding terms

F₈ = F₇ + F₆ = 13 + 8 = 21

Frequently Asked Questions

What's the difference between arithmetic and geometric sequences?

Arithmetic sequences have a constant difference between terms (addition/subtraction), while geometric sequences have a constant ratio between terms (multiplication/division).

How do I identify the pattern in a sequence?

Look at the differences between consecutive terms. If they're constant, it's arithmetic. If the ratios are constant, it's geometric. For more complex patterns, examine second differences or look for polynomial relationships.

Can I create custom formulas?

Yes! Use the custom formula option to create sequences based on any mathematical expression. Use 'n' to represent the term position, and include operations like +, -, *, /, ^, sqrt(), sin(), cos(), etc.

What's the maximum number of terms I can generate?

You can generate up to 1000 terms. For very large numbers or complex calculations, the calculator may limit results to prevent performance issues.