Number Sequence Calculator - Arithmetic, Geometric, and Fibonacci Sequences

Sequence Parameters

Difference between consecutive terms

Generate 1-100 terms

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Explore arithmetic, geometric, and Fibonacci sequences!

Understanding Number Sequences

A number sequence is an ordered list of numbers that follow a specific mathematical pattern or rule. Sequences are fundamental to mathematics and appear in various fields including finance (compound interest), computer science (algorithm analysis), and natural sciences (population growth, radioactive decay).

The three most important types of sequences are arithmetic sequences (where terms differ by a constant), geometric sequences (where terms have a constant ratio), and the Fibonacci sequence (where each term is the sum of the previous two). Understanding these patterns helps solve real-world problems and predict future values.

Types of Number Sequences

Arithmetic Sequence (Linear)

Each term differs from the previous term by a constant value (common difference d).

aā‚™ = a₁ + (n - 1) Ɨ d

Example: 2, 5, 8, 11, 14 (d = 3)

Geometric Sequence (Exponential)

Each term is multiplied by a constant value (common ratio r).

aā‚™ = a₁ Ɨ r^(n-1)

Example: 3, 6, 12, 24, 48 (r = 2)

Fibonacci Sequence

Each term is the sum of the previous two terms, starting with 1, 1.

aā‚™ = aₙ₋₁ + aₙ₋₂

Example: 1, 1, 2, 3, 5, 8, 13, 21, 34

Frequently Asked Questions

What is an arithmetic sequence?

An arithmetic sequence is a sequence where each term differs from the previous term by a constant value called the common difference (d). Formula: aā‚™ = a₁ + (n-1) Ɨ d. Example: 2, 5, 8, 11, 14 has d = 3. The sum of n terms is Sā‚™ = n/2 Ɨ (2a₁ + (n-1)d).

What is a geometric sequence?

A geometric sequence is a sequence where each term is multiplied by a constant value called the common ratio (r). Formula: aā‚™ = a₁ Ɨ r^(n-1). Example: 3, 6, 12, 24 has r = 2. The sum formula is Sā‚™ = a₁ Ɨ (r^n - 1)/(r - 1) when r ≠ 1.

How does the Fibonacci sequence work?

The Fibonacci sequence starts with 1, 1 and each subsequent term is the sum of the previous two terms. Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34. The ratio of consecutive terms converges to the Golden Ratio (ā‰ˆ1.618). It appears frequently in nature, art, and computer science.

How do I find the nth term of a sequence?

Use the general formula for your sequence type. Arithmetic: aā‚™ = a₁ + (n-1) Ɨ d. Geometric: aā‚™ = a₁ Ɨ r^(n-1). Fibonacci: calculate recursively or use Binet's formula. Our calculator computes nth terms instantly for any sequence type.

When does a sequence converge?

Arithmetic sequences never converge (except constant sequences). Geometric sequences converge to 0 when |r| less than 1, and the infinite sum converges to a₁/(1-r). Geometric sequences diverge when |r| ≄ 1. Fibonacci sequence does not converge but the ratio of terms converges to the Golden Ratio.

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