Number Sequence Calculator guide
Number Sequence Calculator: Nth Term and Patterns
Use our number sequence calculator to find the nth term and sum of arithmetic, geometric and Fibonacci sequences, or find the pattern in your numbers.
By the AllCalculator editorial teamUpdated 16 October 20262 min read
A number sequence is a list of numbers that follows a rule. Our Number Sequence Calculator finds any term and the sum, or works out the rule from your numbers.
How to use the calculator (Number Sequence Calculator)
- Choose a sequence type: arithmetic, geometric, Fibonacci, or find the pattern in my numbers.
- Enter the first term, the common difference or ratio and the term number you want.
- Or type your numbers and let the calculator find the rule.
The formulas
| Sequence | nth term | Sum of n terms |
|---|---|---|
| Arithmetic | a + (n − 1)d | n/2 × (first + last) |
| Geometric | a × rⁿ⁻¹ | a(1 − rⁿ) ÷ (1 − r) |
| Fibonacci | Each term is the sum of the two before | — |
A geometric sequence with a ratio between −1 and 1 also has a sum to infinity: a ÷ (1 − r).
A worked example
With the calculator's default settings, an arithmetic sequence starting at 3 with a common difference of 4: 3, 7, 11, 15… The 10th term is 3 + 9 × 4 = 39, and the first 10 terms add up to 210.
Finding the pattern
Type at least 3 numbers and the calculator checks for:
- Arithmetic: the gaps are the same
- Geometric: each term is the previous one times the same number
- Quadratic: the gaps between the gaps are the same, such as 2, 5, 10, 17
- Fibonacci-style: each term is the sum of the two before
Frequently asked questions
What's the nth term?
A formula for any term in the sequence. For 3, 7, 11, 15, it's 4n − 1.
What are triangular and square numbers?
Square numbers (1, 4, 9, 16) and triangular numbers (1, 3, 6, 10) are both quadratic sequences, so the pattern finder spots them.
What's the golden ratio?
In the Fibonacci sequence, each term divided by the one before gets closer and closer to about 1.618, the golden ratio.
Sources and further reading
- GOV.UK: National curriculum in England – mathematicsgov.uk
- Wolfram MathWorld: Sequencemathworld.wolfram.com
Links open official and independent websites in a new tab.
This guide is general information to help you understand your options. It isn't financial, tax, legal or building advice and doesn't take your personal circumstances into account. Figures were checked when the guide was last updated and can change, so confirm anything important with GOV.UK, HMRC, a regulated adviser or a qualified tradesperson.