Number Sequence Calculator
Paste a few terms and find the rule: arithmetic, geometric, quadratic or higher polynomial, or Fibonacci-type, with the formula, the next terms and any term you ask for. Or build an arithmetic or geometric sequence and add it up.
How to Use
- Choose find the rule to paste terms, or arithmetic / geometric to build one.
- Type at least 3 terms separated by commas or spaces (4 or more is safer); fractions and negatives are fine.
- Set n to get that term; for built sequences n is also how many terms to add.
- Read the type, the rule, the next term and the n-th term; the table lists the next five terms and the sums.
- Show Work gives the difference table; the plot shows the terms, with the predicted ones in blue.
Worked Example
2, 6, 12, 20, 30. The differences are 4, 6, 8, 10 and the second differences are all 2, so the rule is quadratic. Fitting it gives aₙ = n² + n (that is, n(n + 1)): the next term is 42 and the 10th is 110.
Gauss’s sum. 1 + 2 + … + 100 is an arithmetic series with a₁ = 1 and a₁₀₀ = 100, so S = 100 × (1 + 100) ÷ 2 = 5,050. Pairing 1 with 100, 2 with 99 and so on gives 50 pairs of 101.
The common mistake: using n instead of n − 1. In 2, 5, 8, 11 the first term already counts as one step, so aₙ = 2 + (n − 1) × 3 = 3n − 1. Writing 2 + 3n gives 5 for the first term, which is wrong.
Show Work
Formulas
Patterns in Numbers
The Rhind papyrus (about 1550 BC) shares loaves in an arithmetic progression, and Euclid’s Elements sums geometric series. Fibonacci’s rabbit problem of 1202 introduced his sequence to Europe, though Indian scholars had found it centuries earlier when counting poetic metres.
The story that the young Carl Friedrich Gauss added 1 to 100 in seconds by pairing terms is told in many versions; the method itself is ancient. The method of finite differences, used here to recognise polynomials, drove Charles Babbage’s Difference Engine, designed in the 1820s to print mathematical tables without human error.
About This Calculator
This calculator recognises arithmetic, geometric, Fibonacci-type and polynomial sequences up to degree 5 from as few as 3 terms, fits the rule with exact fractions, and gives the next five terms and any term up to the 100,000th. In build mode it sums arithmetic and geometric sequences and gives the infinite sum when it converges.
Everything runs in your browser; nothing is sent anywhere.
Related calculators: Exponential Growth Calculator, Zeckendorf Converter, and Data Regression Tool.
Frequently Asked Questions
How do I find the n-th term of a sequence?
Look at the differences between terms. If they are constant the sequence is arithmetic: aₙ = a₁ + (n − 1)d. For 2, 5, 8, 11, d = 3, so aₙ = 3n − 1 and the 100th term is 299. If the ratios are constant it is geometric: aₙ = a₁ · rⁿ⁻¹, so 3, 6, 12, 24 is 3 · 2ⁿ⁻¹.
What if the differences aren’t constant?
Take differences of the differences. Constant second differences mean a quadratic: 2, 6, 12, 20 has differences 4, 6, 8 and second differences 2, so the rule is n² + n. Constant third differences mean a cubic, and so on. The calculator fits the polynomial exactly with Newton’s forward differences.
How do I add up an arithmetic or geometric sequence?
Arithmetic: Sₙ = n(a₁ + aₙ)/2, the average of the first and last term times the number of terms; 1 + 2 + … + 100 = 100 × 101/2 = 5,050. Geometric: Sₙ = a₁(1 − rⁿ)/(1 − r). If |r| < 1 the infinite sum is a₁/(1 − r): 1 + 1/2 + 1/4 + … = 2.
Can a sequence have more than one rule?
Yes. Any finite list fits infinitely many rules: 1, 2, 4 could be doubling (next 8) or n² − n + 2 over 2… (next 7). The calculator gives the simplest rule of the usual kinds that fits every term, so more terms make the answer more certain.
What is a Fibonacci-type sequence?
One where each term is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, … The ratio of consecutive terms approaches the golden ratio 1.618…. Any two starting numbers make one, such as 2, 1, 3, 4, 7, 11 (the Lucas numbers).
How do I use the Number Sequence Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Homework
2, 5, 8, 11 is 3n − 1; the 100th term is 299.
Savings
Saving $50 more each month from $100: month 12 is $650, the total $4,500.
Growth
Doubling from 3: the 10th term is 1,536.
Puzzles
1, 4, 9, 16 → n², next 25.
Series
1 + 1/2 + 1/4 + … adds up to exactly 2.
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