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Nth Term Calculator

Enter the first few terms of a number pattern and the calculator works out the rule. It recognises arithmetic, geometric, quadratic, cubic and higher polynomial sequences plus Fibonacci-style patterns, then gives the nth term formula, the next terms and any term you ask for, with the method of differences shown.

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At least 3 terms, separated by commas or spaces. Decimals and fractions like 3/4 are fine.

Position of the first term
Enter your values to see the result.
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How to use the Nth Term Calculator

  1. Type the terms you know, in order, separated by commas or spaces.
  2. Choose a position n to evaluate, e.g. the 20th term.
  3. Pick the starting position: most textbooks start at n = 1.
  4. Press Find formula to get the nth term rule, next terms and steps.
  5. Check the difference table to see why the pattern fits.

Nth term formulas by sequence type

TypeHow to spot itNth term (first term n = 1)Example
ArithmeticConstant 1st difference daₙ = a₁ + (n − 1)d3, 7, 11, 15 → 4n − 1
GeometricConstant ratio raₙ = a₁ × rn − 13, 6, 12, 24 → 3 × 2n − 1
QuadraticConstant 2nd differenceaₙ = an² + bn + c, a = (2nd diff) ÷ 25, 8, 13, 20 → n² + 4
CubicConstant 3rd differenceaₙ = an³ + bn² + cn + d, a = (3rd diff) ÷ 62, 9, 28, 65 → n³ + 1
Fibonacci-typeEach term = sum of previous twoaₙ = aₙ₋₁ + aₙ₋₂1, 1, 2, 3, 5, 8

The method of differences

Write the terms in a row and subtract each term from the next to get the first differences. Repeat on that row to get the second differences, and so on. If row k becomes constant, the nth term is a polynomial of degree k, and its leading coefficient is the constant difference divided by k! (1 for linear, 2 for quadratic, 6 for cubic, 24 for quartic).

Worked example, 5, 8, 13, 20, 29: first differences 3, 5, 7, 9; second differences 2, 2, 2. The sequence is quadratic and the n² coefficient is 2 ÷ 2 = 1. Subtracting n² (1, 4, 9, 16, 25) leaves 4, 4, 4, 4, 4, so the rule is aₙ = n² + 4 and the 20th term is 404.

Tip: always check your formula by plugging in n = 1 and n = 2. If it reproduces the first terms, it is almost certainly right. The calculator does this check for you and works in exact fractions, so rules like (1/2)n² + (1/2)n for the triangular numbers come out exactly.

Frequently asked questions

How do you find the nth term of a sequence?

Look at the differences between consecutive terms. If the first differences are constant the sequence is arithmetic: aₙ = a₁ + (n − 1)d. If the ratios are constant it is geometric: aₙ = a₁ × r^(n − 1). If the second differences are constant it is quadratic, and so on.

How do you find the nth term of a quadratic sequence?

Halve the constant second difference to get the coefficient of n². Subtract that n² part from each term; what is left is an arithmetic sequence, which gives bn + c. For 5, 8, 13, 20, 29 the second difference is 2, so a = 1; subtracting n² leaves 4, 4, 4…, so aₙ = n² + 4.

How many terms do I need?

Enough to confirm the pattern: at least 3 for an arithmetic or geometric sequence, 4 for a quadratic, 5 for a cubic. With too few terms many rules fit, so the calculator only reports a pattern it has seen repeat.

What is the difference between arithmetic and geometric sequences?

An arithmetic sequence adds the same number each time (3, 7, 11, 15: +4). A geometric sequence multiplies by the same number each time (3, 6, 12, 24: ×2). Arithmetic sequences grow linearly; geometric ones grow or shrink exponentially.

Should the first term be n = 1 or n = 0?

Most school courses number the first term n = 1. Some programming and textbook problems start at n = 0. The formula changes (for 3, 7, 11 it is 4n − 1 starting at 1, or 4n + 3 starting at 0), so pick the convention your question uses.

Why does it say no pattern was found?

The terms may follow a rule that is not polynomial, geometric or Fibonacci-like (for example primes or alternating signs mixed with growth), there may be a typo, or there are not enough terms to confirm a higher-degree pattern. Adding one or two more terms often solves it.

Are all the tools on FreeToolCabin tested for quality and accuracy?

Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.

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