Understanding Number Sequences

A number sequence is a list of numbers arranged in a specific order, following a particular rule or pattern. This calculator focuses on two of the most common types: arithmetic and geometric sequences.

Arithmetic Sequences

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant value is called the common difference (d).

Example: 5, 8, 11, 14, 17, ... (Here, the common difference is 3).

Formulas:

• Nth Term (aₙ): `aₙ = a₁ + (n - 1) * d`
• Sum of First n Terms (Sₙ): `Sₙ = (n / 2) * (2a₁ + (n - 1) * d)`

Geometric Sequences

A geometric sequence is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

Example: 2, 6, 18, 54, 162, ... (Here, the common ratio is 3).

Formulas:

• Nth Term (aₙ): `aₙ = a₁ * r^(n - 1)`
• Sum of First n Terms (Sₙ): `Sₙ = a₁ * (1 - rⁿ) / (1 - r)` (for r ≠ 1)

Real-World Applications

• Finance: Geometric sequences are the foundation of compound interest calculations. Arithmetic sequences can model simple interest or regular savings plans. Some of these can be explored with our Financial Calculators.
• Physics: Modeling radioactive decay (geometric) or an object's position under constant acceleration (arithmetic).
• Biology: Predicting population growth under certain conditions (often geometric).
• Computer Science: Analyzing the complexity of some algorithms.

As sequences can grow very large, you might find our Big Number Calculator useful for handling the results.

Related Math Tools

• Math Calculators - Our full suite of mathematical utilities.
• Rounding Calculator - For rounding your final results to a desired precision.
• Scientific Notation Calculator - For working with very large or small numbers.