Number Sequence Calculator
Nth term and series sum for arithmetic, geometric and Fibonacci sequences with up to 20 visible terms.
Inputs
Nth term
39.00000000
Series sum
210.00000000
Sequence (first 10 terms)
3, 7, 11, 15, 19, 23, 27, 31, 35, 39
Step by step
Sequence type
= Arithmetic
First term a₁
= 3
Common difference d
= 4
Term 10
a₁ + (n−1)d = 3 + (10−1) × 4
= 39
Sum of first 10 terms
10/2 × (2×3 + (10−1)×4)
= 210
First 10 terms
= 3, 7, 11, 15, 19, 23, 27, 31, 35, 39
How it works
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.
Formulas
Arithmetic nth term
aₙ = a₁ + (n−1)d
- a₁
- First term
- d
- Common difference
- n
- Term index
Arithmetic series sum
Sₙ = n/2 × (2a₁ + (n−1)d)
Geometric nth term
aₙ = a₁ × r^(n−1)
- r
- Common ratio
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
Frequently Asked Questions
What is the difference between a sequence and a series?
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).
Does the Fibonacci sequence start at 0 or 1?
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.
Can an infinite geometric series converge?
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.