How do I find the sum of the first n terms?

For an arithmetic sequence, Sn = n(a1 + an)/2. For a geometric sequence with r not equal to 1, Sn = a1(1 − r^n)/(1 − r). When r equals 1, Sn = n times a1.

Can the common difference or ratio be negative or zero?

Yes. A negative arithmetic difference makes terms decrease. A negative geometric ratio makes signs alternate. A zero ratio makes every geometric term after the first equal to zero.

Does the calculator store my sequence values?

No. The sequence calculation runs locally in the browser, and the calculator does not upload or store the entered values.

Arithmetic and Geometric Sequence Calculator

Choose a sequence type and enter its first term, common difference or ratio, and term number. All calculations run locally in your browser; the calculator does not upload or store your values.

Find the nth term, the sum of the first n terms, the sequence formula, a generated term list, and worked substitutions.

Enter the sequence

Sequence result

Nth term

a₁₀ = 39

Arithmetic sequence with common difference 4.

Finite sumS₁₀ = 210

Explicit ruleaₙ = 3 + (n − 1) · 4

Terms3, 7, 11, 15, 19, 23, 27, 31, 35, 39

Step-by-step calculation

Arithmetic and geometric sequence formulas

Arithmetic nth term

aₙ = a₁ + (n − 1)d

Add the common difference d once for every step after the first term.

Arithmetic finite sum

Sₙ = n(a₁ + aₙ) / 2

The average of the first and last terms, multiplied by the number of terms.

Geometric nth term

aₙ = a₁rⁿ⁻¹

Multiply by the common ratio r for each step after the first term.

Geometric finite sum

Sₙ = a₁(1 − rⁿ) / (1 − r)

This form applies when r ≠ 1. If r = 1, then S_n = na₁.

How to identify the sequence type

Check differences

Subtract each term from the next. If the difference stays constant, the sequence is arithmetic. For 5, 8, 11, 14, the common difference is 3.

Check ratios

Divide each nonzero term by the previous term. If the ratio stays constant, the sequence is geometric. For 2, 6, 18, 54, the common ratio is 3.

Negative values

A negative difference usually produces a decreasing arithmetic sequence. A negative ratio makes a geometric sequence alternate signs.

Sequence versus series

A sequence is an ordered list of terms. A series is the sum of those terms. This calculator gives both a_n and the finite sum S_n.

Accuracy, privacy, and limits

The calculator uses standard real-number formulas and browser floating-point arithmetic, displaying up to 12 significant digits. For a positive geometric ratio extremely close to 1, it uses logarithmic functions designed to reduce cancellation in the finite-sum calculation.

Inputs are limited to finite values with absolute value at most 10¹⁰⁰ and n from 1 to 100,000. If a requested term or sum exceeds the browser’s finite numeric range, the calculator reports an error. At most 50 terms are displayed to keep the page responsive; the requested nth term and sum still use the full value of n. Last reviewed: 4 August 2026.

Sequence calculator FAQs

What is the difference between an arithmetic and geometric sequence?

An arithmetic sequence adds the same difference each time. A geometric sequence multiplies by the same ratio each time.

What is the nth-term formula for an arithmetic sequence?

Use aₙ = a₁ + (n − 1)d, where a₁ is the first term, d is the common difference, and n is the term position.

What is the nth-term formula for a geometric sequence?

Use aₙ = a₁r^(n−1), where r is the common ratio. The exponent counts how many times the first term is multiplied by r.

How is the finite sum calculated?

For an arithmetic sequence, use Sₙ = n(a₁ + aₙ)/2. For a geometric sequence with r ≠ 1, use Sₙ = a₁(1 − rⁿ)/(1 − r). When r = 1, every term equals the first term, so Sₙ = na₁.

Can the difference or ratio be zero or negative?

Yes. A difference of zero makes a constant arithmetic sequence. A ratio of zero makes all geometric terms after the first zero. A negative ratio alternates signs.

Why are only 50 terms displayed?

The display is capped so very large requests do not freeze the page or create an unwieldy result. The calculator still computes the requested nth term and sum for the full value of n.