How to use the number sequence calculator
- Choose the sequence type: arithmetic, geometric, Fibonacci-like, or find the pattern if you have a list of numbers and don't know the rule.
- Enter the starting values: the first term and the common difference or ratio, the first two terms for a Fibonacci-like sequence, or at least three terms of your list.
- Enter the term number n. The calculator gives the nth term and the sum of the first n terms.
- Read the rule, the step-by-step working and a table of the first terms (up to 100) with a running total.
Sequence formulas
a₁ is the first term and n is the position of the term you want. In an arithmetic sequence you add the common difference d each time, so the terms grow in a straight line. In a geometric sequence you multiply by the common ratio r each time, so they grow (or shrink) by the same percentage. A Fibonacci-like sequence starts from any two numbers and each new term is the sum of the two before it. Sₙ is the sum of the first n terms.
When the ratio of a geometric sequence is between −1 and 1, the terms shrink toward zero and the sum settles on a limit, a₁ ÷ (1 − r). The calculator shows that "sum to infinity" when it exists.
Worked example
The calculator opens with an arithmetic sequence that starts at 3 and adds 4 each time: 3, 7, 11, 15, and so on.
- The 10th term: a₁₀ = 3 + (10 − 1) × 4 = 3 + 36 = 39.
- The sum of the first 10 terms: S₁₀ = 10 × (3 + 39) ÷ 2 = 10 × 21 = 210.
- The rule in simplest form: aₙ = 3 + 4(n − 1) = 4n − 1. Try n = 10: 40 − 1 = 39.
Switch to geometric with a₁ = 3 and r = 2 and the 10th term is 3 × 2⁹ = 3 × 512 = 1,536, with a sum of 3 × (2¹⁰ − 1) = 3,069. The classic Fibonacci sequence 1, 1, 2, 3, 5, 8, … reaches 55 at the 10th term, and its first 10 terms add up to 143.
How the pattern finder works
Paste three or more numbers and the calculator tests them in this order:
| Test | Example | Result |
|---|---|---|
| Differences all equal | 5, 8, 11, 14 | Arithmetic, d = 3 |
| Ratios all equal | 2, 6, 18, 54 | Geometric, r = 3, next term 162 |
| Each term = sum of the previous two | 2, 5, 7, 12, 19 | Fibonacci-like |
| Differences of the differences equal | 1, 4, 9, 16, 25 | Quadratic, aₙ = n² |
The default list, 2, 6, 18, 54, has ratios of 3 at every step, so the next term is 54 × 3 = 162. If none of the tests pass, the calculator says so instead of guessing. Any finite list fits infinitely many rules, so more terms make the answer more trustworthy.
Common mistakes
- Using n instead of n − 1. The first term already counts as one step, so the 10th term of an arithmetic sequence adds the difference 9 times, not 10.
- Mixing up difference and ratio. 2, 4, 6, 8 adds 2 each time; 2, 4, 8, 16 doubles each time. They agree for two terms and then split apart quickly.
- Forgetting negative ratios. With r = −2 the terms alternate in sign: 3, −6, 12, −24.
Frequently asked questions
What is the difference between a sequence and a series?
A sequence is the list of terms, such as 3, 7, 11, 15. A series is what you get when you add them up, 3 + 7 + 11 + 15 = 36. This calculator gives both: the nth term of the sequence and the sum of its first n terms.
How do I find the common difference?
Subtract any term from the one after it. In 3, 7, 11, 15 the difference is 7 − 3 = 4. If the gaps are not all the same, the sequence is not arithmetic; try the pattern finder.
Why does the Fibonacci ratio approach 1.618?
Divide any Fibonacci term by the one before it and the answer gets closer and closer to the golden ratio, (1 + √5) ÷ 2 ≈ 1.618034. This happens for any Fibonacci-like sequence that starts with two positive numbers, which the working shows for your inputs.
Can the terms be decimals or negative?
Yes. The first term, the difference, the ratio and the starting values can all be decimals or negative numbers. Only the term number n has to be a whole number of 1 or more.
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