About the number sequence calculator
This calculator builds arithmetic, geometric and Fibonacci-style sequences from a first term and a rule, then gives the nth term, the sum of the first n terms and a table of the terms. For geometric sequences it also reports the sum to infinity when it exists.
It is designed for checking homework and exploring how sequences grow.
Who should use the number sequence calculator
- Algebra and pre-calculus students.
- Teachers preparing examples.
- Anyone modelling steady additions or percentage growth over steps.
How to use the number sequence calculator: what each input means
- Sequence type
- Arithmetic (constant difference), geometric (constant ratio) or Fibonacci-style (each term the sum of the previous two).
- First term
- The starting value a₁.
- Common difference or ratio
- d for arithmetic and r for geometric sequences.
- Second term
- For Fibonacci-style sequences, the second starting value.
- Number of terms
- Whole number from 1 to 1,000.
How the number sequence calculator works: method and formula
In an arithmetic sequence each term adds the common difference, so the nth term is a₁ + (n − 1)d. In a geometric sequence each term multiplies by the ratio, so the nth term is a₁ × r^(n−1). Sums use closed-form formulas rather than adding terms one by one, except in Fibonacci-style sequences, where terms are generated and added.
- Arithmetic term
- aₙ = a₁ + (n − 1)d
- Arithmetic sum
- Sₙ = n(a₁ + aₙ) ÷ 2
- Geometric term
- aₙ = a₁ × r^(n−1)
- Geometric sum
- Sₙ = a₁(1 − rⁿ) ÷ (1 − r), S∞ = a₁ ÷ (1 − r) when |r| < 1
How to read your number sequence calculator result
The infinite sum of a geometric sequence exists only when the ratio is strictly between −1 and 1; otherwise the sum does not converge. Values beyond about 9 × 10¹⁵ are shown rounded because ordinary decimal numbers cannot hold every digit.
Number Sequence Calculator worked example
An arithmetic sequence starting at 3 with a difference of 4, ten terms; a geometric one starting at 2 with ratio 3; and the Fibonacci sequence.
- Arithmetic terms: 3, 7, 11, 15, 19, 23, 27, 31, 35, 39.
- The 10th term is 3 + 9 × 4 = 39, and the sum is 10 × (3 + 39) ÷ 2 = 210.
- Geometric (2, ratio 3) first six terms: 2, 6, 18, 54, 162, 486.
- Fibonacci-style with a₁ = a₂ = 1, first ten terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.
The 10th term of the arithmetic example is 39 and the sum of the first ten terms is 210.
Practical uses of the number sequence calculator
- Checking nth-term and sum formulas.
- Modelling regular savings (arithmetic) or percentage growth (geometric).
- Exploring Fibonacci-like patterns with different starting values.
Assumptions and limitations of this calculator
- Up to 1,000 terms and double-precision numbers, so very large terms are approximated.
- Does not find the rule from a list of terms; enter the first term and the difference or ratio.