Sequence & Series
Arithmetic Sequence Calculator
Find the nth term, the sum of the first n terms and list an arithmetic sequence from its first term and common difference.
aₙ = a₁ + (n − 1)d
Also called arithmetic progression · ap calculator
Math · 11 calculators
Sequence & Series Calculators
Arithmetic and geometric sequences, nth terms, partial and infinite sums, Fibonacci and harmonic sequences.
Sequence & Series
Geometric Sequence Calculator
Compute the nth term and partial sum of a geometric sequence from the first term and common ratio, and list the terms.
aₙ = a₁·rⁿ⁻¹
Also called geometric progression · gp calculator
Sequence & Series
Fibonacci Calculator
Find the nth Fibonacci number exactly (up to n = 5,000) and list the sequence, with the golden-ratio approximation.
F(0) = 0, F(1) = 1, F(n) = F(n−1) + F(n−2)
Also called fibonacci sequence · nth fibonacci number
Sequence & Series
Arithmetic Series Calculator
Sum an arithmetic series from its first and last terms or first term and common difference, with the number of terms.
S = n(a₁ + aₙ)/2, n = (aₙ − a₁)/d + 1
Also called sum of arithmetic series · arithmetic sum
Sequence & Series
Geometric Series Calculator
Calculate the sum of a finite geometric series, and whether it converges as n → ∞ (|r| < 1).
Sₙ = a₁(1 − rⁿ)/(1 − r)
Also called sum of geometric series · geometric sum
Sequence & Series
Sequence Sum Calculator
Sum of the first n natural numbers, squares, cubes, odd or even numbers using closed-form formulas.
Σk = n(n+1)/2
Also called sum of natural numbers · sum of squares
Sequence & Series
Nth Term Calculator
Paste the first few terms of a sequence; the calculator detects arithmetic, geometric or quadratic patterns and finds the nth term formula.
Arithmetic: a + (n−1)d
Also called find nth term · sequence rule
Sequence & Series
Harmonic Sequence Calculator
Generate a harmonic sequence (reciprocals of an arithmetic sequence), its nth term and partial sum.
hₙ = 1 / (a₁ + (n − 1)d)
Also called harmonic progression · hp calculator
Sequence & Series
Infinite Geometric Series Calculator
Find the sum to infinity of a geometric series and check convergence, e.g. 1 + 1/2 + 1/4 + … = 2.
S∞ = a₁ / (1 − r), valid for |r| < 1
Also called sum to infinity · infinite sum
Sequence & Series
Common Difference Calculator
Find the common difference of an arithmetic sequence from a list of terms or from two terms and their positions.
d = (aₙ − aₘ) / (n − m)
Also called find common difference · arithmetic difference
Sequence & Series
Common Ratio Calculator
Find the common ratio of a geometric sequence from two terms and their positions, plus the first term.
r = (aₙ / aₘ)^(1/(n − m))
Also called find common ratio · geometric ratio
Key sequence & series formulas
| Arithmetic Sequence | aₙ = a₁ + (n − 1)d · Sₙ = n/2 × (a₁ + aₙ) |
|---|---|
| Geometric Sequence | aₙ = a₁·rⁿ⁻¹ · Sₙ = a₁(1 − rⁿ)/(1 − r) |
| Fibonacci | F(0) = 0, F(1) = 1, F(n) = F(n−1) + F(n−2) |
| Arithmetic Series | S = n(a₁ + aₙ)/2, n = (aₙ − a₁)/d + 1 |
| Geometric Series | Sₙ = a₁(1 − rⁿ)/(1 − r) · S∞ = a₁/(1 − r) for |r| < 1 |
| Sequence Sum | Σk = n(n+1)/2 · Σk² = n(n+1)(2n+1)/6 · Σk³ = [n(n+1)/2]² |
| Nth Term | Arithmetic: a + (n−1)d · Geometric: a·rⁿ⁻¹ · Quadratic: an² + bn + c |
| Harmonic Sequence | hₙ = 1 / (a₁ + (n − 1)d) |
| Infinite Geometric Series | S∞ = a₁ / (1 − r), valid for |r| < 1 |
| Common Difference | d = (aₙ − aₘ) / (n − m) |
Sequence & Series questions
How is Fibonacci related to the golden ratio?
The ratio of consecutive Fibonacci numbers F(n+1)/F(n) approaches the golden ratio φ ≈ 1.618 as n grows.