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Sequence & Series Calculator

Generate terms, find the nth term formula, compute partial sums and check convergence for arithmetic, geometric, Fibonacci, harmonic and power series. Includes sum of natural numbers, squares and cubes.

Arithmetic & GeometricConvergence Check
Arithmetic Sequence
Geometric Sequence
Fibonacci Sequence
Harmonic Series
Power Series Expansion
Custom Sequence Pattern Detection

Arithmetic vs Geometric Sequences

The two most fundamental types of sequences in mathematics differ in how each term relates to the previous one.

PropertyArithmeticGeometric
PatternAdd constant d each stepMultiply by constant r each step
nth termaₙ = a₁ + (n−1)daₙ = a₁ × r^(n−1)
Finite sum Sₙn/2 × (2a₁ + (n−1)d)a₁(1 − rⁿ)/(1 − r), r ≠ 1
Infinite sum S∞Diverges (unless d=0)a₁/(1 − r) if |r| < 1
Mean of n termsArithmetic mean = (a₁ + aₙ)/2Geometric mean = (a₁ × aₙ)^(1/2)
Example1, 4, 7, 10, 13, … (d=3)2, 6, 18, 54, … (r=3)

Convergence and Divergence of Series

An infinite series converges if its partial sums approach a finite limit. Key tests for convergence:

Famous Sequences in Mathematics

Several sequences appear repeatedly across mathematics, nature, and science:

Frequently Asked Questions

A sequence is an ordered list of numbers following a pattern, such as 1, 3, 5, 7, 9, … A series is the sum of the terms of a sequence: 1 + 3 + 5 + 7 + 9 + … = 25. Sequences describe individual terms; series describe their cumulative total. A series can be finite (summing a fixed number of terms) or infinite (which may converge to a finite value or diverge to infinity).
An infinite geometric series a + ar + ar² + ar³ + … converges to a/(1−r) if and only if |r| < 1. If |r| ≥ 1, the series diverges. For example, 1 + 1/2 + 1/4 + 1/8 + … converges to 2, while 1 + 2 + 4 + 8 + … diverges to infinity.
The Fibonacci sequence starts with F₁=1, F₂=1, and each subsequent term is the sum of the previous two: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … The ratio of consecutive Fibonacci numbers converges to the golden ratio φ ≈ 1.61803. Binet's formula gives the nth term directly: Fₙ = (φⁿ − ψⁿ)/√5, where ψ = (1−√5)/2.
For an arithmetic sequence: aₙ = a₁ + (n−1)d. For a geometric sequence: aₙ = a₁ × r^(n−1). For quadratic sequences, fit aₙ = an² + bn + c using the method of finite differences. The Custom Sequence tab of this calculator automatically detects the pattern type from the first few terms and provides the formula.