Sequence Calculator

Q: How do I find the rule of a sequence?

Take the differences of neighbouring terms. If they are all equal, the sequence is arithmetic. If the ratios of neighbouring terms are equal, it is geometric. If only the differences of the differences are equal, it is quadratic.

Q: How do I find the nth term?

For an arithmetic sequence aₙ = a₁ + (n − 1)·d, for a geometric one aₙ = a₁·rⁿ⁻¹. For 3, 7, 11 with d = 4 the tenth term is 3 + 9·4 = 39.

Q: How do I sum the first n terms?

Arithmetic: Sₙ = n·(a₁ + aₙ) ÷ 2, the classic Gauss formula. Geometric: Sₙ = a₁·(rⁿ − 1) ÷ (r − 1). For 3, 7, 11 … the first ten terms add up to 210.

Q: What is the Fibonacci sequence?

A sequence in which every term is the sum of the previous two: 1, 1, 2, 3, 5, 8, 13 … The ratio of neighbouring terms approaches the golden ratio of about 1.618.

Q: Why does the calculator sometimes find no rule?

Only the four sequence types above are checked. Others, such as the primes, follow different rules. With more terms the detection becomes more reliable.

This sequence calculator finds the rule behind a list of numbers, continues it and gives the formula for the nth term. It also adds up the first n terms – or builds an arithmetic or geometric sequence from a first term and a step.

Source: Wikipedia – Arithmetic progression (formula and sum). Updated: 1 October 2026.

What do you want to do?

Sequence

Separate numbers with commas, spaces or line breaks. At least 3 numbers.

for the nth term and the sum of the first n terms (1 to 200)

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Result

Type of sequence: Arithmetic sequence with common difference d = 4
Formula: aₙ = 3 + (n − 1) · 4 = 4n − 1
The next 5 terms: 23, 27, 31, 35, 39
nth term: a10 = 39
Sum of the first n terms: S10 = 210
Sum to infinity
Terms and sums: n Term aₙ Sum Sₙ
1 3 3
2 7 10
3 11 21
4 15 36
5 19 55
6 23 78
7 27 105
8 31 136
9 35 171
10 39 210; A rule inferred from a few terms is only a suggestion: infinitely many formulas fit any finite list. The calculator reports the simplest rule it recognizes.

n	Term aₙ	Sum Sₙ
1	3	3
2	7	10
3	11	21
4	15	36
5	19	55
6	23	78
7	27	105
8	31	136
9	35	171
10	39	210

How it is calculated

A sequence is an ordered list of numbers a₁, a₂, a₃ … that follows a fixed rule. The calculator checks whether the terms have a constant difference, a constant ratio, the Fibonacci rule or a constant second difference.

The four sequences it recognizes

Arithmetic sequence: each term differs from the previous one by the same d. Formula aₙ = a₁ + (n − 1)·d, sum of the first n terms Sₙ = n·(a₁ + aₙ) ÷ 2. Example 3, 7, 11, 15: d = 4.
Geometric sequence: each term is r times the previous one. Formula aₙ = a₁·rⁿ⁻¹, sum Sₙ = a₁·(rⁿ − 1) ÷ (r − 1). Example 2, 6, 18, 54: r = 3. If |r| < 1, the infinite series approaches a₁ ÷ (1 − r).
Fibonacci-type sequence: each term is the sum of the previous two, such as 1, 1, 2, 3, 5, 8.
Quadratic sequence: the differences between terms form an arithmetic sequence, so the second difference is constant. Example 1, 4, 9, 16 with aₙ = n².

Where this is used

Arithmetic sequences appear in steady instalment plans, geometric ones in growth and interest – see the compound interest calculator and the exponent calculator. For the mean of a list of values, use the average calculator.

Frequently asked questions

How do I find the rule of a sequence?

Take the differences of neighbouring terms. If they are all equal, the sequence is arithmetic. If the ratios of neighbouring terms are equal, it is geometric. If only the differences of the differences are equal, it is quadratic.

How do I find the nth term?

For an arithmetic sequence aₙ = a₁ + (n − 1)·d, for a geometric one aₙ = a₁·rⁿ⁻¹. For 3, 7, 11 with d = 4 the tenth term is 3 + 9·4 = 39.

How do I sum the first n terms?

Arithmetic: Sₙ = n·(a₁ + aₙ) ÷ 2, the classic Gauss formula. Geometric: Sₙ = a₁·(rⁿ − 1) ÷ (r − 1). For 3, 7, 11 … the first ten terms add up to 210.

What is the Fibonacci sequence?

A sequence in which every term is the sum of the previous two: 1, 1, 2, 3, 5, 8, 13 … The ratio of neighbouring terms approaches the golden ratio of about 1.618.

Why does the calculator sometimes find no rule?