Use this free series calculator to solve arithmetic and geometric sequences, find nth terms and finite sums, and check the formula behind each result. You can also explore our growing library of practical calculators for finance, business, salary, home projects and more.
A sequence lists terms in order; a series adds those terms together. This calculator handles both ideas. Choose an arithmetic pattern with a constant difference or a geometric pattern with a constant ratio, then calculate the nth term and the sum of the first n terms.
2, 5, 8, 11, 14, 17, 20, 23
Sₙ = n/2 × [2a₁ + (n − 1)d]
For an arithmetic sequence, each term changes by the same common difference.
An arithmetic sequence changes by the same amount each step. If the first term is 2 and the common difference is 3, the sequence is 2, 5, 8, 11, ... .
A geometric sequence multiplies by the same ratio each step. If the first term is 3 and the common ratio is 2, the sequence is 3, 6, 12, 24, ... .
A mathematical sequence is an ordered list of terms. A series is the result of adding terms from a sequence. In everyday searches, the two ideas are often grouped together, so this series calculator shows both the requested term and the finite sum.
You may see the same kind of tool described as a series calc, series math calculator, calculator for series, or series solver. Here, the focus is on the two common patterns that can be solved directly from a first term and either a difference or ratio.
| Series type | How terms change | nth term | Finite sum |
|---|---|---|---|
| Arithmetic | Add the same difference d | an = a1 + (n − 1)d | Sn = n/2 × [2a1 + (n − 1)d] |
| Geometric | Multiply by the same ratio r | an = a1rn−1 | Sn = a1(1 − rn) / (1 − r) |
An infinite geometric series has a finite sum only when the absolute value of the common ratio is less than 1. In that case, the sum is S∞ = a1 / (1 − r). If |r| is 1 or greater, the terms do not shrink toward zero fast enough for the infinite sum to converge.
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Use these quick answers for the core concepts, then use the calculator above to check a specific arithmetic or geometric pattern.
Geometric sequences are useful when a quantity changes by the same factor or percentage over equal steps. Examples include compound growth, depreciation, some population models, repeated bouncing where each bounce reaches a fixed fraction of the previous height, and other growth or decay patterns. The model is appropriate only when the ratio is reasonably constant for the situation being studied.
Start with the first term a1 and multiply by the common ratio r to generate each next term. To jump directly to the nth term, use an = a1rn−1. For example, if a1 = 3 and r = 2, the first four terms are 3, 6, 12, and 24.
Choose the geometric option, enter the first term, common ratio, and number of terms, then calculate. For a finite geometric series with r ≠ 1, the formula is Sn = a1(1 − rn) / (1 − r). When |r| < 1, the calculator also shows the convergent infinite sum S∞ = a1 / (1 − r).
An arithmetic sequence has a constant difference between consecutive terms, so it changes by repeated addition or subtraction. A geometric sequence has a constant ratio, so it changes by repeated multiplication or division.
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Series Calculator combines fast calculator interfaces with readable explanations. When a topic needs additional care, review the calculator's methodology, source notes and disclaimer before using the result for a real-world decision.
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