Sequence Calculator: Arithmetic, Geometric, nth Term & Sum

A sequence calculator finds the rule behind a list of numbers, then gives any term and the sum. For 7, 11, 15, 19 the difference is always 4, so the sequence is arithmetic: aₙ = 7 + (n − 1) · 4 = 4n + 3. The 20th term is 83, and the first 20 terms add up to 900.

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How to use the sequence calculator

1. Find the rule: type at least three terms (four or more for curved patterns) and choose n. The calculator tests for arithmetic, geometric, quadratic, cubic, Fibonacci-type and “multiply then add” rules, then shows the nth term formula, the next five terms, aₙ and the sum of the first n terms.
2. Arithmetic or Geometric: enter the first term, the common difference d or ratio r, and n. You get aₙ, the sum Sₙ, the formula, the recursive rule and a table of terms. Fractions such as 1/2 and negative numbers work.
3. Fibonacci: choose the two starting terms (1, 1 by default) and n, up to the 5,000th term with every digit exact.

Each result has a bar chart of the first terms and step-by-step working you can open.

Types of sequences at a glance

Arithmetic sequence formulas

An arithmetic sequence adds the same number d (the common difference) each time. With first term a₁:

• nth term: aₙ = a₁ + (n − 1)d
• Sum of the first n terms: Sₙ = n/2 · (a₁ + aₙ) = n/2 · (2a₁ + (n − 1)d)

Worked example: 7, 11, 15, 19, … has a₁ = 7 and d = 4. The 20th term is 7 + 19 · 4 = 83. The sum of the first 20 terms is 20/2 · (7 + 83) = 10 · 90 = 900. Simplifying the rule gives aₙ = 4n + 3, a straight-line pattern with slope 4.

The sum formula is Gauss’s pairing trick: the first and last terms, the second and second-to-last, and so on all have the same total, so 1 + 2 + … + 100 = 50 · 101 = 5,050.

Geometric sequence formulas

A geometric sequence multiplies by the same number r (the common ratio) each time:

• nth term: aₙ = a₁ · rⁿ⁻¹
• Sum of the first n terms: Sₙ = a₁(1 − rⁿ) / (1 − r), for r ≠ 1 (if r = 1, Sₙ = n · a₁)
• Infinite sum: S∞ = a₁ / (1 − r), only when −1 < r < 1

Worked example: 5, 10, 20, 40, … has r = 2. The 8th term is 5 · 2⁷ = 640 and the first 8 terms add up to 5(1 − 2⁸) / (1 − 2) = 1,275. For 4, 12, 36, 108, … the ratio is 3, so aₙ = 4 · 3ⁿ⁻¹. For 100, 50, 25, … the ratio is 1/2, the terms shrink toward 0, and the infinite sum is 100 / (1 − 1/2) = 200.

A negative ratio makes the signs alternate: 81, −27, 9, −3 has r = −1/3.

Quadratic sequences and the difference method

When the first differences are not constant, take differences again. If the second differences are constant, the sequence is quadratic, and the n² coefficient is half the second difference.

The second difference is 2, so a = 1. Subtracting n² from 2, 5, 10 leaves 1, 1, 1, so aₙ = n² + 1 and the 20th term is 401. The triangular numbers 1, 3, 6, 10, 15 work the same way: the second difference is 1, so aₙ = n(n + 1)/2 and the 20th is 210. Constant third differences mean a cubic, such as the cubes 1, 8, 27, 64.

The Fibonacci sequence

The Fibonacci sequence starts 1, 1 and each term is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, … The 20th term is 6,765. A handy identity gives the sum without adding everything: a₁ + a₂ + … + aₙ = aₙ₊₂ − a₂, so the first 20 Fibonacci numbers add up to 17,711 − 1 = 17,710.

The ratio of neighbouring terms approaches the golden ratio φ = (1 + √5)/2 ≈ 1.618. Some books start at F₀ = 0, F₁ = 1, which shifts the numbering by one; set the first two terms to 0 and 1 to match. Starting with 2 and 1 gives the Lucas numbers 2, 1, 3, 4, 7, 11, …

How to find the next number in a sequence

1. Subtract neighbouring terms. A constant difference means arithmetic: 3, 8, 13, 18 → 23.
2. Divide neighbouring terms. A constant ratio means geometric: 2, 6, 18, 54 → 162.
3. Take second differences. Constant means quadratic: 1, 4, 9, 16 → 25 (the square numbers).
4. Check whether each term is the sum of the two before (Fibonacci-type) or a multiple of the previous term plus a constant: 3, 7, 15, 31 → 63 (double and add 1).

A short list can fit more than one rule, so use as many terms as you have. Two terms are never enough to fix a pattern.

Sequences and series

A sequence is the list of terms; a series is their sum. Sₙ above is a partial sum, written with sigma notation as Σ aₖ from k = 1 to n. Arithmetic series always grow without limit (unless every term is 0), while a geometric series with |r| < 1 settles on the finite value a₁ / (1 − r).

Common mistakes and how to check an answer

A quick habit: after you find a rule, substitute n = 1 and n = 2 and compare with the given terms, then estimate whether the size of your answer makes sense (a doubling sequence grows very fast; an arithmetic one grows steadily).

Sequences practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you find the nth term of a sequence?

Decide whether the terms add or multiply by a constant. If they add d, use aₙ = a₁ + (n − 1)d; if they multiply by r, use aₙ = a₁ · rⁿ⁻¹. If the second differences are constant, fit aₙ = an² + bn + c.

How do you know if a sequence is arithmetic?

Subtract each term from the next. If every difference is the same, the sequence is arithmetic. 7, 11, 15, 19 has differences 4, 4, 4.

What is the formula for a geometric sequence?

The nth term is aₙ = a₁ · rⁿ⁻¹, and the sum of the first n terms is a₁(1 − rⁿ) / (1 − r) when r ≠ 1.

How do you find the formula for a sequence from its terms?

Test the differences, then the ratios, then the second differences. Write the matching rule and check it with n = 1 and n = 2. The Find the rule tab does all three tests at once.

How do you find the next number in a sequence?

Find the constant amount added or multiplied, or the constant second difference, and apply it once more. 3, 8, 13, 18 is followed by 23; 2, 6, 18, 54 by 162.

When does an infinite geometric series have a sum?

Only when the ratio is between −1 and 1. Then the sum is a₁ / (1 − r); for 100, 50, 25, … it is 200.

Related calculators

• Fibonacci calculator: Fibonacci numbers, the golden ratio and Binet's formula.
• Summation calculator: evaluate sigma notation for any series.
• Exponential growth calculator: geometric growth over time.
• Compound interest calculator: a geometric sequence with money.
• Slope calculator: the common difference is the slope of an arithmetic sequence.