Using the Sequence Calculator
Select Arithmetic for a constant difference or Geometric for a constant ratio. Then choose the input mode that matches the information you have.
- Choose First term & common difference/ratio, Two known terms, or List with missing values.
- Enter the fields shown for that mode. If needed, enter a value in Optional: also solve for term n and S_n.
- Click Calculate to show the sequence values and checks.
- Review Formula & steps, then use Copy result or Download result (.txt).
Choose an Input Mode
- First term & common difference/ratio: enter
a_1anddfor an arithmetic sequence, ora_1andrfor a geometric sequence. - Two known terms: enter two different positive whole-number positions and their values. The calculator derives
dorr, then works back toa_1. - List with missing values: enter comma-separated terms. Mark a gap with
_,?, an empty entry, or the implementation-supportedx/Xmarker. At least two known terms are required.
Choose the sequence type yourself. The calculator does not automatically select arithmetic or geometric.
Optional Term and Sum
Optional: also solve for term n and S_n is needed only when you want an additional term or the finite sum of the first n terms. Enter a valid positive whole-number target; without it, a_n and S_n are not calculated. Two-known-term positions and the target cannot exceed 1,000,000,000.
What the Result Shows
The result lists a_1 and d or r. When you enter n, it also lists a_n and S_n. A geometric sequence with |r| < 1 also receives an infinite sum, S_∞.
Formula & steps shows the calculation. The result also includes a sequence preview or a filled sequence. A normal preview shows at most 50 terms; when a requested n is larger, the preview is truncated. Non-whole values are displayed rounded to at most six visible decimal places after the internal calculation step.
Formulas and Indexing
Positions are 1-based: the first term is n = 1.
- Arithmetic:
a_n = a_1 + (n - 1) × dandS_n = n / 2 × (2 × a_1 + (n - 1) × d). - Geometric:
a_n = a_1 × r^(n - 1). Forr ≠ 1,S_n = a_1 × (1 - r^n) / (1 - r); forr = 1, the calculator usesS_n = a_1 × n. - For
|r| < 1, the infinite sum isS_∞ = a_1 / (1 - r).
Validation and Limitations
- Number fields accept finite decimal-style values. The parser does not accept every textual form of a real number, such as scientific notation or a plus-prefixed value.
- For geometric calculations, the first term and ratio cannot be zero. Two-known-term solving can fail when no real common ratio satisfies the supplied positions and values.
- In list mode, the calculator derives a strict model from at least two known terms, checks every supplied term, and reports mismatches instead of forcing a result.
- It does not infer arbitrary, nonlinear, recursive, Fibonacci, squared, or other rules. It tests only the selected constant-difference or constant-ratio model.
- Values are treated as plain numbers. Units are not inferred, labeled, or converted.
- The tool is not a substitute for professional or certified calculations.
Calculation Privacy Scope
The calculator logic runs in your browser and does not transmit the terms you enter. This describes the calculator logic only; it does not establish that no page-level analytics or other page behavior occurs.
Frequently Asked Questions
What is the difference between an arithmetic and a geometric sequence?
In an arithmetic sequence, each term after the first is found by adding a fixed common difference (d) to the term before it. In a geometric sequence, each term after the first is found by multiplying the term before it by a fixed common ratio (r).
How does the calculator fill in missing values?
You enter the sequence with blanks (using _, ?, an empty spot between commas, or x/X) for the unknown terms and pick arithmetic or geometric. The calculator needs at least two known terms to derive the common difference or ratio, uses that to compute the first term, checks every other known term against the model, and only then fills the blanks.
What happens if my supplied terms do not fit the selected model?
The calculator computes the common difference or ratio from two known terms and checks every other supplied term against it. If any term disagrees with the model beyond rounding, it reports exactly which position does not fit instead of guessing a different rule or forcing a result.
Can this tool find the rule for any sequence, like a Fibonacci or squared sequence?
No. This tool only tests and computes strict arithmetic (constant difference) and geometric (constant ratio) models. It will not guess a nonlinear or other rule from your terms, and it will not claim a unique rule from a single known value - it needs at least two known terms and reports when the data doesn't fit the model you selected.
What indexing convention does this calculator use?
Positions are 1-based: the first term you list or supply is position n = 1, the second is n = 2, and so on. This is stated alongside every result so it isn't ambiguous.