Number Sequence Calculator
Arithmetic & Geometric Sequence Tool
Solve for the nth term and the sum of the first n terms of a sequence you define.
Paste at least three terms, separated by commas or spaces, to auto-detect the sequence type.
Understanding Number Sequences
A number sequence is an ordered list of numbers following a rule. The two most common patterns — arithmetic and geometric — describe an enormous range of real-world growth, from simple counting to compound interest.
Each term is the previous term plus a fixed common difference (d). Arithmetic sequences grow (or shrink) at a constant, linear rate — like a savings account with fixed monthly deposits, or seats numbered down a row.
Each term is the previous term multiplied by a fixed common ratio (r). Geometric sequences grow (or decay) exponentially — like compound interest, population growth, or radioactive decay.
The Formulas
Arithmetic
Geometric
Detecting a Sequence's Pattern
If you already have a list of numbers and want to know what kind of sequence they form, the Detect tab compares the differences and ratios between consecutive terms. If every consecutive difference is the same (within a small tolerance for rounding), the list is arithmetic; if every consecutive ratio is the same, it's geometric. Detection needs at least three terms — two numbers alone don't contain enough information to distinguish a pattern — and terms that mix both behaviors, or don't follow either rule, are reported as not a simple arithmetic or geometric sequence.
The constant added between terms in an arithmetic sequence.
The constant multiplier between terms in a geometric sequence.
At least three terms are required to reliably auto-detect a sequence's type.
Key Takeaways
- Arithmetic sequences grow linearly — each term adds the same fixed amount (d) to the previous one.
- Geometric sequences grow exponentially — each term multiplies the previous one by the same fixed ratio (r).
- The sum formulas are closed-form — you don't need to add up every term by hand to find the total of the first n terms.
- Detection has limits. It needs at least three terms and only recognizes simple arithmetic or geometric patterns, not more complex sequences.
Frequently Asked Questions
- On the Solve tab, pick Arithmetic or Geometric from the type selector.
- Enter the first term (a₁) and either the common difference (d) or common ratio (r), whichever field is shown.
- Enter which term you want to solve for (n).
- Click "Calculate" to see the nth term and the sum of the first n terms.
An arithmetic sequence is a list of numbers where each term is found by adding the same fixed amount — the common difference (d) — to the previous term. For example, 2, 5, 8, 11, 14 is arithmetic with d = 3, since each term is 3 more than the one before it.
A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same fixed amount — the common ratio (r). For example, 3, 6, 12, 24, 48 is geometric with r = 2, since each term is double the one before it.
The Detect tab checks whether the consecutive differences between your terms are all equal (arithmetic) or whether the consecutive ratios between your terms are all equal (geometric), allowing a small tolerance for floating-point rounding error. It needs at least three terms to make a meaningful comparison — with only two terms, any pair of numbers can trivially be described as either type. If neither pattern fits within tolerance, or your terms mix arithmetic and geometric behavior, it reports that the sequence isn't a simple arithmetic or geometric sequence.
For an arithmetic sequence: the nth term is a_n = a1 + (n-1)d, and the sum of the first n terms is S_n = n/2 × (2a1 + (n-1)d). For a geometric sequence: the nth term is a_n = a1 × r^(n-1), and the sum of the first n terms is S_n = a1 × (1 - r^n) / (1 - r) (or simply a1 × n when r = 1).
The value of n tells the calculator how far along the sequence to go. Both the nth-term and sum formulas are closed-form expressions that depend directly on n, so changing n changes both the value of that specific term and how many terms are included in the running total.