Find terms and sums of arithmetic, geometric and Fibonacci sequences.
An arithmetic sequence adds a fixed number (common difference) each step: 3, 7, 11, 15 … A geometric sequence multiplies by a fixed ratio each step: 2, 6, 18, 54 … The Fibonacci sequence adds the two previous terms: 1, 1, 2, 3, 5, 8, 13 … Each has closed-form formulas for the nth term and the sum of the first n terms.
Arithmetic nth term
aₙ = a₁ + (n−1)d
Arithmetic series sum
Sₙ = n/2 × (2a₁ + (n−1)d)
Geometric nth term
aₙ = a₁ × r^(n−1)
Geometric series sum
Sₙ = a₁ × (1 − rⁿ) / (1 − r) for r ≠ 1
Fibonacci sum identity
Sum of first n Fibonacci numbers = F(n+2) − 1
A sequence is an ordered list of numbers (e.g. 3, 7, 11, 15). A series is the sum of the terms in a sequence (e.g. 3 + 7 + 11 + 15 = 36).
Conventions differ. This calculator uses the most common school convention: F₁ = 1, F₂ = 1, so the sequence starts 1, 1, 2, 3, 5 … The alternative starting at 0 shifts all indices by one.
Yes — when |r| < 1. The infinite sum is a₁ / (1 − r). For example, 1 + 0.5 + 0.25 + … converges to 1 / (1 − 0.5) = 2. This calculator computes finite sums; use the limit formula for the infinite case.