Education

Nth Term Calculator

Enter a sequence and find the nth term rule. The calculator checks arithmetic, geometric, and quadratic patterns, then shows the formula, differences, generated terms, and the requested term.

nth-term-calculator
Enter at least the first 4 terms if you want the calculator to detect arithmetic, geometric, or quadratic patterns. Use commas between terms.
Nth term result
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Detected pattern—
Formula—
First differences—
Second differences—
Generated terms—
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What does “nth term” mean?

The nth term is the rule that tells you the value of a term from its position number. Instead of listing every term one by one, an nth-term formula lets you jump directly to the 20th, 50th, or 100th term. This is why students search for an nth term calculator: they are not only trying to continue a pattern, they are trying to find a general rule.

For arithmetic patterns, OpenStax uses the explicit formula aₙ = a₁ + (n - 1)d in its arithmetic sequences lesson. For geometric patterns, the explicit formula uses powers of the common ratio, as shown in OpenStax’s geometric sequences lesson.

How the calculator chooses the rule

The calculator starts by checking first differences. If those are constant, the sequence is arithmetic. If not, it checks ratios. If the ratios are constant, the sequence is geometric. If neither works, it checks second differences, which is the common classroom test for a quadratic sequence. This is not a full symbolic pattern solver for every possible sequence in mathematics, but it covers the patterns students meet most often in algebra.

Arithmetic: constant first difference Geometric: constant ratio Quadratic: constant second difference

Examples of nth-term patterns

TermsPattern testTypeTypical nth term
4, 9, 14, 19+5 each timeArithmeticaₙ = 4 + 5(n - 1)
3, 6, 12, 24×2 each timeGeometricaₙ = 3 × 2ⁿ⁻¹
2, 5, 10, 17second difference = 2Quadraticaₙ = n² + 1

Why some sequences cannot be solved from a few terms

A short list of numbers may fit more than one possible rule. For example, 2, 4, 8 could be geometric doubling, but another more complicated formula could also begin with those same three values. In school problems, the expected rule is usually the simplest one that matches the topic being studied. That is why this calculator clearly says which type of pattern it detected instead of pretending every sequence has one obvious answer.

Worked example: quadratic nth term

Sequence: 2, 5, 10, 17, 26 First differences: 3, 5, 7, 9 Second differences: 2, 2, 2 Because the second difference is constant, the rule is quadratic. If aₙ = An² + Bn + C, then 2A = 2, so A = 1. Solving from the first terms gives B = 0 and C = 1. nth term: aₙ = n² + 1

Common questions

  • Start by looking at the differences between consecutive terms. If the differences are the same, use the arithmetic formula. If the differences are not the same, check whether each term is multiplied by the same ratio to get the next term. If neither of those works, check the second differences. This simple order of testing catches most school-level nth-term questions without guessing.
  • An arithmetic nth-term rule has the form a_n = a_1 + (n - 1)d. It is used when the same number is added or subtracted each time. For example, 7, 11, 15, 19 has a common difference of 4, so the rule is a_n = 7 + 4(n - 1), which can also be simplified to a_n = 4n + 3.
  • A geometric nth-term rule has the form a_n = a_1r^(n - 1). It is used when the same multiplier connects each term. For example, 5, 10, 20, 40 has first term 5 and common ratio 2, so the rule is a_n = 5 × 2^(n - 1). The exponent is n - 1 because the first term has not been multiplied yet.
  • Check the second differences. If the second differences are constant, the sequence is quadratic and the rule has the form a_n = An^2 + Bn + C. The constant second difference equals 2A, so A is half the second difference. Then the first few terms can be used to solve for B and C. This calculator does that algebra automatically for simple quadratic sequences.
  • The terms may not form an arithmetic, geometric, or quadratic sequence, or there may not be enough terms to identify a reliable rule. Some sequences are based on cubes, factorials, alternating rules, prime numbers, or a written context. A calculator can test common patterns, but it cannot always know the intended rule from a short list of numbers alone.
  • Yes. A limited list of terms does not always prove a unique rule. Many different formulas can be built to match the same first few values. In classroom algebra, the expected answer is usually the simplest pattern that matches the lesson, such as arithmetic, geometric, or quadratic. That is why showing the pattern test is important, not just the final formula.
  • Usually yes, especially if your teacher expects a simplified expression. For example, a_n = 4 + 5(n - 1) is correct, but it simplifies to a_n = 5n - 1. The unsimplified version is often easier to understand because it shows the first term and common difference. The simplified version is often cleaner for final answers.
  • Most school sequence problems start at n = 1, where a_1 is the first term. Some applications, especially in finance, computer science, or recursive formulas, may start at n = 0. This calculator assumes the first term you enter is a_1. If your problem starts at n = 0, adjust the indexing or use the rule carefully.