Number Sequence Calculator
Find terms and sums of arithmetic, geometric and Fibonacci sequences.
About Number Sequence Calculator
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
Worked example
With the default values below, Number Sequence Calculator returns Nth term of 39.00000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Sequence typeone of Arithmetic (add a constant difference), Geometric (multiply by a constant ratio), Fibonacci (each term = sum of two previous)
- Arithmetic (add a constant difference)
- First term (a₁)
- 3
- Common difference (d)
- 4
- Common ratio (r)
- 2
- Find the nth term (n)minimum 1
- 10
- Sum first how many terms?accepts 1–1,000
- 10
Results
- 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
- 1Sequence type= Arithmetic
- 2First term a₁= 3
- 3Common difference d= 4
- 4Term 10= 39
- 5Sum of first 10 terms= 210
- 6First 10 terms= 3, 7, 11, 15, 19, 23, 27, 31, 35, 39
Assumptions and limits
- The result is derived only from the 6 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.